20:
394:
7353:, who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable. Earlier authors held the result to be intuitively obvious and requiring no proof. The insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of
7348:
as an example) by providing an algorithm for constructing the decimal expansion of the solution. The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration. Before the formal definition of continuity was given, the
8279:
The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of continuous, real-valued functions of a real variable, to continuous functions in general spaces.
3572:
2632:
10002:
4237:
4946:
7437:. The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.
4741:
4548:
4422:
4999:
939:
8511:
8126:
4477:
9420:
7155:
7199:
6154:
3638:
6673:
6527:
5967:
4598:
8393:
753:
6747:
6113:
4346:
4099:
3371:
2431:
4795:
1393:
5515:
4649:
8059:
6039:
3400:
2460:
6596:
5780:
9359:
8774:
5371:
5321:
5168:
5118:
5025:
4023:
3019:
2079:
1713:
1494:
143:
7303:
6451:
5891:
4831:
3758:
2813:
8920:
3683:
2738:
8222:
7965:
4140:
3441:
2501:
9002:
8718:
8359:
6477:
5917:
3143:
2203:
719:
6814:
6221:
5634:
5271:
5221:
1643:
1211:
8958:
8549:
7991:
7071:
6375:
4278:
3959:
3791:
3716:
2846:
2771:
1466:
977:
9666:
shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.
9604:
9333:
9298:
8809:
6942:
6782:
6708:
6410:
6189:
6074:
5850:
5692:
5550:
3826:
3123:
2881:
2183:
1809:
7326:
3446:
2506:
1941:
1335:
1156:
9228:
7239:
7091:
3921:
3895:
8870:
8844:
8584:
7259:
7111:
7012:
6977:
6880:
6313:
6247:
5815:
5576:
5406:
5060:
3994:
1619:
1564:
1529:
1237:
1012:
562:
527:
492:
240:
5444:
4310:
1774:
8674:
8275:
8024:
5724:
3261:
3192:
3048:
2993:
2321:
2252:
2108:
2053:
1295:
1266:
1050:
838:
809:
356:
327:
7040:. Bryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area. The theorem was first proved by
588:
8639:
8337:
697:
9662:
9560:
9540:
9520:
9500:
9473:
9453:
9379:
9184:
9164:
8413:
8242:
8190:
8170:
7929:
7909:
7219:
6907:
6854:
6834:
6340:
6287:
6267:
5654:
5596:
4871:
4851:
4669:
4063:
4043:
3866:
3846:
3321:
3301:
3281:
3232:
3212:
3163:
3088:
3068:
2964:
2944:
2921:
2901:
2381:
2361:
2341:
2292:
2272:
2223:
2148:
2128:
2024:
2004:
1981:
1961:
1909:
1889:
1869:
1849:
1829:
1736:
1584:
780:
648:
628:
608:
424:
272:
205:
185:
164:
93:
41:
9263:
2698:
457:
73:
7524:
Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions; this definition was not adopted.
4876:
4674:
10190:
4482:
9669:
The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily met constraints).
7541:
is a generalization of the
Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an
4951:
4145:
843:
7864:
7332:
provided the modern formulation and a proof in 1821. Both were inspired by the goal of formalizing the analysis of functions and the work of
650:. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.
10175:
7373:
that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values
7502:
tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the
4351:
10280:
9983:
9502:
to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points
10119:
9762:
8418:
1340:
8064:
4427:
9613:
will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which
9904:
10290:
9384:
10012:
7116:
7357:
in Cauchy's case and using real inequalities in
Bolzano's case), and to provide a proof based on such definitions.
7160:
1412:
10191:
https://mathoverflow.net/questions/253059/approximate-intermediate-value-theorem-in-pure-constructive-mathematics
7510:
6118:
3577:
2637:
6601:
6482:
5922:
4553:
10221:
10202:
10135:
Matthew Frank (July 14, 2020). "Interpolating
Between Choices for the Approximate Intermediate Value Theorem".
9843:. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 23–36.
8364:
724:
6713:
6079:
4315:
4068:
3329:
2389:
10285:
7349:
intermediate value property was given as part of the definition of a continuous function. Proponents include
4746:
9948:(2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography.
5449:
4603:
8036:
5972:
3376:
2436:
9127:
7518:
6532:
5729:
1659:
7538:
9841:
Redefining
Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction
9338:
8729:
5326:
5276:
5123:
5073:
5004:
4002:
2998:
2058:
1668:
1471:
98:
10068:
10029:
7264:
6415:
5855:
4800:
3721:
2776:
9130:
is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.
8875:
8590:
The intermediate value theorem is an immediate consequence of these two properties of connectedness:
3643:
8195:
7938:
7336:. The idea that continuous functions possess the intermediate value property has an earlier origin.
9988:
9139:
7880:
4104:
3405:
2465:
9978:
9687: – A measurable set with positive measure that contains no subset of smaller positive measure
8963:
8679:
8342:
7883:
and follows from the basic properties of connected sets in metric spaces and connected subsets of
6456:
5896:
3128:
2188:
702:
19:
10069:"Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros"
9432:
7503:
6787:
6194:
5601:
5226:
5176:
1624:
1175:
8925:
8516:
7970:
7050:
6345:
4245:
3926:
3763:
3688:
2818:
2743:
1429:
944:
9949:
9565:
9303:
9268:
8779:
6912:
6752:
6678:
6380:
6159:
6044:
5820:
5662:
5520:
3796:
3093:
2851:
2153:
1779:
286:
278:
10232:
7311:
1917:
1300:
1055:
9189:
7333:
7329:
7224:
7076:
7021:
3900:
3874:
247:
8849:
8814:
8554:
7244:
7096:
6982:
6947:
6859:
6292:
6226:
5785:
5555:
5376:
5030:
3964:
1589:
1534:
1499:
1216:
982:
532:
497:
462:
210:
9856:
8137:
5411:
4283:
1741:
282:
9681: – On the existence of a tangent to an arc parallel to the line through its endpoints
8650:
8251:
8000:
5700:
3237:
3168:
3024:
2969:
2297:
2228:
2084:
2029:
1271:
1242:
1026:
814:
785:
332:
303:
8:
10265:
9974:
9142:, the intermediate value theorem is not true. Instead, one has to weaken the conclusion:
7722:
7484:
7037:
1021:
567:
381:
275:
8606:
8304:
664:
10208:
10144:
9927:
9819:
9768:
9699:
9693:
9684:
9678:
9647:
9545:
9525:
9505:
9485:
9458:
9438:
9364:
9169:
9149:
8398:
8227:
8175:
8155:
7914:
7894:
7204:
7033:
6892:
6839:
6819:
6325:
6272:
6252:
5639:
5581:
4856:
4836:
4654:
4048:
4028:
3851:
3831:
3306:
3286:
3266:
3217:
3197:
3148:
3073:
3053:
2949:
2929:
2906:
2886:
2366:
2346:
2326:
2277:
2257:
2208:
2133:
2113:
2009:
1989:
1966:
1946:
1894:
1874:
1854:
1834:
1814:
1721:
1569:
1423:
765:
633:
613:
593:
409:
370:
369:
If a continuous function has values of opposite sign inside an interval, then it has a
257:
190:
170:
149:
78:
26:
9236:
8960:
must actually hold, and the desired conclusion follows. The same argument applies if
7024:, which places "intuitive" arguments involving infinitesimals on a rigorous footing.
430:
46:
16:
Continuous function on an interval takes on every value between its values at the ends
10229:
10217:
10115:
10090:
10049:
10008:
9888:
9871:
9772:
9758:
9724:
9721:
9120:
8147:
7032:
A form of the theorem was postulated as early as the 5th century BCE, in the work of
1169:
10158:
7308:
The equivalence between this formulation and the modern one can be shown by setting
10154:
10080:
10041:
9953:
9919:
9883:
9844:
9750:
7366:
9839:
Bos, Henk J. M. (2001). "The legitimation of geometrical procedures before 1590".
9852:
9231:
8245:
7994:
7041:
3567:{\displaystyle |x-b|<\delta \implies |f(x)-f(b)|<f(b)-u\implies f(x)>u.}
2627:{\displaystyle |x-a|<\delta \implies |f(x)-f(a)|<u-f(a)\implies f(x)<u.}
1416:
10085:
10045:
9848:
10253:
9028:
8283:
Recall the first version of the intermediate value theorem, stated previously:
7350:
7345:
9961:
9957:
9754:
10274:
10094:
10053:
7354:
7552:
Vrahatis presents a similar generalization to triangles, or more generally,
10261:
10212:
10171:
9945:
9643:
In general, for any continuous function whose domain is some closed convex
7932:
7337:
393:
9011:
The intermediate value theorem generalizes in a natural way: Suppose that
10248:
10109:
9024:
402:
9475:-space will always map some pair of antipodal points to the same place.
4232:{\displaystyle |x-c|<\delta _{1}\implies |f(x)-f(c)|<\varepsilon }
1963:
is the smallest number that is greater than or equal to every member of
9931:
9905:"Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus"
7514:
7341:
7020:
The intermediate value theorem can also be proved using the methods of
10237:
9729:
401:
This captures an intuitive property of continuous functions over the
362:
9923:
9872:"A translation of Bolzano's paper on the intermediate value theorem"
10227:
10149:
9824:
7876:
5657:
1912:
4941:{\displaystyle f(c)>f(a^{**})-\varepsilon \geq u-\varepsilon .}
1395:
A subset of the real numbers with no internal gap is an interval.
9818:
Sanders, Sam (2017). "Nonstandard
Analysis and Constructivism!".
8061:
is connected if and only if it satisfies the following property:
7557:
7044:
in 1817. Bolzano used the following formulation of the theorem:
4736:{\displaystyle f(c)<f(a^{*})+\varepsilon <u+\varepsilon .}
384:
of a continuous function over an interval is itself an interval.
9005:
7867:. In can be used for approximations of fixed points and zeros.
7546:
9719:
4543:{\displaystyle f(x)-\varepsilon <f(c)<f(x)+\varepsilon }
3125:
is a strict inequality, consider the similar implication when
4417:{\displaystyle (c-\delta _{2},c+\delta _{2})\subseteq (a,b)}
10249:"Two-dimensional version of the Intermediate Value Theorem"
1172:
of function values has no gap. For any two function values
7875:
The intermediate value theorem is closely linked to the
7870:
4994:{\displaystyle u-\varepsilon <f(c)<u+\varepsilon }
934:{\displaystyle \min(f(a),f(b))<u<\max(f(a),f(b)),}
9689:
Pages displaying short descriptions of redirect targets
8506:{\displaystyle \min(f(a),f(b))<u<\max(f(a),f(b))}
6944:
is impossible. If we combine both results, we get that
4600:. By the properties of the supremum, there exists some
1415:. The intermediate value theorem does not apply to the
10266:
http://mizar.org/version/current/html/topreal5.html#T4
2185:
is a strict inequality, consider the implication when
10030:"Generalization of the Bolzano theorem for simplices"
9650:
9568:
9548:
9528:
9508:
9488:
9461:
9441:
9387:
9367:
9341:
9306:
9271:
9239:
9192:
9172:
9152:
8966:
8928:
8878:
8852:
8817:
8782:
8732:
8682:
8653:
8609:
8557:
8519:
8421:
8401:
8367:
8345:
8307:
8254:
8230:
8198:
8178:
8158:
8121:{\displaystyle x,y\in E,\ x<r<y\implies r\in E}
8067:
8039:
8003:
7973:
7941:
7917:
7897:
7314:
7267:
7247:
7227:
7207:
7163:
7119:
7099:
7079:
7053:
6985:
6950:
6915:
6895:
6862:
6842:
6822:
6790:
6755:
6716:
6681:
6604:
6535:
6485:
6459:
6418:
6383:
6348:
6328:
6295:
6275:
6255:
6229:
6197:
6162:
6121:
6082:
6047:
5975:
5925:
5899:
5858:
5823:
5788:
5732:
5703:
5665:
5642:
5604:
5584:
5558:
5523:
5452:
5446:, which is more intuitive. We further define the set
5414:
5379:
5329:
5279:
5229:
5179:
5126:
5076:
5033:
5007:
4954:
4879:
4859:
4839:
4803:
4749:
4677:
4657:
4606:
4556:
4485:
4472:{\displaystyle \delta =\min(\delta _{1},\delta _{2})}
4430:
4354:
4318:
4286:
4248:
4148:
4107:
4071:
4051:
4031:
4005:
3967:
3929:
3903:
3877:
3854:
3834:
3799:
3766:
3724:
3691:
3646:
3580:
3449:
3408:
3379:
3332:
3309:
3289:
3269:
3240:
3220:
3200:
3171:
3151:
3131:
3096:
3076:
3056:
3027:
3001:
2972:
2952:
2932:
2909:
2889:
2854:
2821:
2779:
2746:
2701:
2640:
2509:
2468:
2439:
2392:
2369:
2349:
2329:
2300:
2280:
2260:
2231:
2211:
2191:
2156:
2136:
2116:
2087:
2061:
2032:
2012:
1992:
1969:
1949:
1920:
1897:
1877:
1857:
1837:
1817:
1782:
1744:
1724:
1671:
1627:
1592:
1572:
1537:
1502:
1474:
1432:
1343:
1303:
1274:
1245:
1219:
1178:
1058:
1029:
985:
947:
846:
817:
788:
768:
727:
705:
667:
658:
The intermediate value theorem states the following:
636:
616:
596:
570:
535:
500:
465:
433:
412:
335:
306:
260:
213:
193:
173:
152:
101:
81:
49:
29:
9113:. The original theorem is recovered by noting that
8676:, is also connected. For convenience, assume that
9656:
9598:
9554:
9534:
9514:
9494:
9467:
9447:
9414:
9373:
9353:
9327:
9292:
9257:
9222:
9178:
9158:
8996:
8952:
8914:
8864:
8838:
8803:
8768:
8712:
8668:
8633:
8578:
8543:
8505:
8407:
8387:
8353:
8331:
8269:
8236:
8216:
8184:
8164:
8120:
8053:
8018:
7985:
7959:
7923:
7903:
7320:
7297:
7253:
7233:
7213:
7193:
7149:
7105:
7085:
7065:
7006:
6971:
6936:
6901:
6874:
6848:
6828:
6808:
6776:
6741:
6702:
6667:
6590:
6521:
6471:
6445:
6404:
6369:
6334:
6307:
6281:
6261:
6241:
6215:
6183:
6148:
6107:
6068:
6033:
5961:
5911:
5885:
5844:
5809:
5774:
5718:
5686:
5656:is bounded and non-empty, so by Completeness, the
5648:
5628:
5590:
5570:
5544:
5509:
5438:
5400:
5365:
5315:
5265:
5215:
5162:
5112:
5054:
5019:
4993:
4940:
4865:
4845:
4825:
4789:
4735:
4663:
4643:
4592:
4542:
4471:
4416:
4340:
4304:
4272:
4231:
4134:
4093:
4057:
4037:
4017:
3988:
3953:
3915:
3889:
3860:
3840:
3820:
3785:
3752:
3710:
3677:
3632:
3566:
3435:
3394:
3365:
3315:
3295:
3275:
3255:
3226:
3206:
3186:
3157:
3137:
3117:
3082:
3062:
3042:
3013:
2987:
2958:
2938:
2915:
2895:
2875:
2840:
2807:
2765:
2732:
2692:
2626:
2495:
2454:
2425:
2375:
2355:
2335:
2315:
2286:
2266:
2246:
2217:
2197:
2177:
2142:
2122:
2102:
2073:
2047:
2018:
1998:
1975:
1955:
1935:
1903:
1883:
1863:
1843:
1823:
1803:
1768:
1730:
1707:
1658:The theorem may be proven as a consequence of the
1637:
1613:
1578:
1558:
1523:
1488:
1460:
1411:The theorem depends on, and is equivalent to, the
1387:
1329:
1289:
1260:
1231:
1205:
1150:
1044:
1006:
971:
933:
832:
803:
774:
747:
713:
691:
642:
622:
602:
582:
556:
521:
486:
451:
418:
350:
321:
266:
234:
199:
179:
158:
137:
87:
67:
35:
9972:
7517:of some other function on some interval have the
10272:
9702: – Theorem on triangulation graph colorings
9606:. If the line is rotated 180 degrees, the value
9415:{\displaystyle \vert f(x)\vert <\varepsilon }
8467:
8422:
7073:be continuous functions on the interval between
5672:
4437:
3584:
2650:
1927:
1239:, even if they are outside the interval between
1105:
1066:
892:
847:
7513:states that all functions that result from the
7150:{\displaystyle f(\alpha )<\varphi (\alpha )}
9133:
10134:
7194:{\displaystyle f(\beta )>\varphi (\beta )}
5852:. Then, by the definition of continuity, for
1422:because gaps exist between rational numbers;
1362:
1346:
1322:
1306:
1143:
1061:
10209:Intermediate value Theorem - Bolzano Theorem
9435:, which says that a continuous map from the
9403:
9388:
9230:be a pointwise continuous function from the
5504:
5459:
3283:, which means there are values smaller than
2343:, which means there are values greater than
7532:
7521:(even though they need not be continuous).
6149:{\displaystyle N>{\frac {\delta }{b-c}}}
3633:{\displaystyle (\max(a,b-\delta ),b]=I_{2}}
2693:{\displaystyle [a,\min(a+\delta ,b))=I_{1}}
1426:fill those gaps. For example, the function
1406:
1052:is also a closed interval, and it contains
300:, then it takes on any given value between
10114:. New York: McGraw-Hill. pp. 42, 93.
8108:
8104:
7340:proved the intermediate value theorem for
6668:{\displaystyle -g(c)<g(x)-g(c)<g(c)}
6522:{\displaystyle x\in (c-\delta ,c+\delta )}
5962:{\displaystyle x\in (c-\delta ,c+\delta )}
4593:{\displaystyle x\in (c-\delta ,c+\delta )}
4185:
4181:
3542:
3538:
3479:
3475:
2602:
2598:
2539:
2535:
10148:
10084:
10000:
9887:
9823:
9426:
8388:{\displaystyle f\colon I\to \mathbb {R} }
8381:
8347:
8047:
7612:, that never equals 0 on the boundary of
5817:. For contradiction, let us assume, that
1662:property of the real numbers as follows:
1482:
748:{\displaystyle f\colon I\to \mathbb {R} }
741:
707:
10066:
10027:
9902:
9696: – Theorem in differential topology
6742:{\displaystyle x=c-{\frac {\delta }{2}}}
6108:{\displaystyle x=c+{\frac {\delta }{N}}}
4341:{\displaystyle \exists \delta _{2}>0}
4094:{\displaystyle \exists \delta _{1}>0}
3366:{\displaystyle \varepsilon =f(b)-u>0}
3323:. A more detailed proof goes like this:
2426:{\displaystyle \varepsilon =u-f(a)>0}
2383:. A more detailed proof goes like this:
392:
18:
9984:MacTutor History of Mathematics Archive
9896:
9817:
9791:. Appleton-Century-Crofts. p. 284.
7863:The theorem can be proved based on the
5062:as the only possible value, as stated.
4790:{\displaystyle a^{**}\in (c,c+\delta )}
1566:. However, there is no rational number
1388:{\displaystyle {\bigl }\subseteq f(I).}
10273:
9786:
7765:; then the conditions become simpler:
7706:on all points on the face opposite to
7328:to the appropriate constant function.
5510:{\displaystyle S=\{x\in :g(x)\leq 0\}}
4644:{\displaystyle a^{*}\in (c-\delta ,c]}
564:must pass through the horizontal line
10228:
10107:
9802:
9744:
9720:
9015:is a connected topological space and
8641:is a connected set. It follows from
8054:{\displaystyle E\subset \mathbb {R} }
7871:General metric and topological spaces
7865:Knaster–Kuratowski–Mazurkiewicz lemma
7476:. This function is not continuous at
10246:
9869:
7620:satisfies the following conditions:
7360:
6034:{\displaystyle |g(x)-g(c)|<-g(c)}
3395:{\displaystyle \exists \delta >0}
2455:{\displaystyle \exists \delta >0}
1986:Note that, due to the continuity of
43:be a continuous function defined on
10137:Logical Methods in Computer Science
10111:Principles of Mathematical Analysis
10067:Vrahatis, Michael N. (2020-04-15).
10028:Vrahatis, Michael N. (2016-04-01).
9838:
9265:to the real line, and suppose that
7692:is not equal to the sign-vector of
7014:is the only remaining possibility.
6591:{\displaystyle |g(x)-g(c)|<g(c)}
5775:{\displaystyle g(c)<0,g(c)>0}
5697:There are 3 cases for the value of
2926:Likewise, due to the continuity of
358:at some point within the interval.
13:
10184:
9903:Grabiner, Judith V. (March 1983).
9119:is connected and that its natural
7527:
5065:
4319:
4108:
4072:
3409:
3380:
2469:
2440:
1891:is non-empty and bounded above by
1653:
14:
10302:
10196:
9912:The American Mathematical Monthly
9354:{\displaystyle \varepsilon >0}
9335:. Then for every positive number
8769:{\displaystyle f(a)<u<f(b)}
7440:As an example, take the function
5366:{\displaystyle g(a)<0<g(b)}
5316:{\displaystyle f(a)<u<f(b)}
5163:{\displaystyle f(a)>u>f(b)}
5113:{\displaystyle f(a)<u<f(b)}
5020:{\displaystyle \varepsilon >0}
4018:{\displaystyle \varepsilon >0}
3014:{\displaystyle \varepsilon >0}
2074:{\displaystyle \varepsilon >0}
1708:{\displaystyle f(a)<u<f(b)}
1489:{\displaystyle x\in \mathbb {Q} }
138:{\displaystyle f(a)<s<f(b)}
7740:It is possible to normalize the
7604:) be a continuous function from
7298:{\displaystyle f(x)=\varphi (x)}
6446:{\displaystyle \epsilon =g(c)-0}
5886:{\displaystyle \epsilon =0-g(c)}
4826:{\displaystyle a^{**}\not \in S}
3753:{\displaystyle |x-b|<\delta }
2808:{\displaystyle |x-a|<\delta }
1413:completeness of the real numbers
23:Intermediate value theorem: Let
10176:How to stabilize a wobbly table
10165:
10128:
10101:
10060:
10021:
10001:Smorynski, Craig (2017-04-07).
9994:
9966:
9381:in the unit interval such that
8915:{\displaystyle u\neq f(a),f(b)}
5070:We will only prove the case of
3678:{\displaystyle I_{2}\subseteq }
2733:{\displaystyle I_{1}\subseteq }
1665:We shall prove the first case,
10281:Theory of continuous functions
10222:Wolfram Demonstrations Project
9938:
9863:
9832:
9811:
9795:
9779:
9738:
9713:
9593:
9587:
9578:
9572:
9400:
9394:
9322:
9316:
9281:
9275:
9252:
9240:
9214:
9211:
9199:
8991:
8985:
8976:
8970:
8947:
8935:
8909:
8903:
8894:
8888:
8827:
8821:
8798:
8792:
8763:
8757:
8742:
8736:
8707:
8701:
8692:
8686:
8663:
8657:
8628:
8616:
8567:
8561:
8538:
8526:
8500:
8497:
8491:
8482:
8476:
8470:
8455:
8452:
8446:
8437:
8431:
8425:
8377:
8326:
8314:
8264:
8258:
8217:{\displaystyle f\colon X\to Y}
8208:
8105:
8013:
8007:
7960:{\displaystyle f\colon X\to Y}
7951:
7292:
7286:
7277:
7271:
7188:
7182:
7173:
7167:
7144:
7138:
7129:
7123:
6995:
6989:
6960:
6954:
6925:
6919:
6765:
6759:
6691:
6685:
6662:
6656:
6647:
6641:
6632:
6626:
6617:
6611:
6585:
6579:
6569:
6565:
6559:
6550:
6544:
6537:
6516:
6492:
6453:and know, that there exists a
6434:
6428:
6393:
6387:
6358:
6352:
6172:
6166:
6057:
6051:
6028:
6022:
6009:
6005:
5999:
5990:
5984:
5977:
5956:
5932:
5880:
5874:
5833:
5827:
5798:
5792:
5763:
5757:
5742:
5736:
5713:
5707:
5681:
5675:
5623:
5611:
5533:
5527:
5495:
5489:
5480:
5468:
5433:
5421:
5389:
5383:
5360:
5354:
5339:
5333:
5310:
5304:
5289:
5283:
5254:
5248:
5239:
5233:
5204:
5198:
5189:
5183:
5157:
5151:
5136:
5130:
5107:
5101:
5086:
5080:
5043:
5037:
4976:
4970:
4914:
4898:
4889:
4883:
4784:
4766:
4709:
4696:
4687:
4681:
4638:
4620:
4587:
4563:
4531:
4525:
4516:
4510:
4495:
4489:
4466:
4440:
4411:
4399:
4393:
4355:
4299:
4287:
4267:
4255:
4219:
4215:
4209:
4200:
4194:
4187:
4182:
4164:
4150:
4129:
4117:
3977:
3971:
3948:
3936:
3809:
3803:
3740:
3726:
3672:
3660:
3614:
3605:
3587:
3581:
3552:
3546:
3539:
3529:
3523:
3513:
3509:
3503:
3494:
3488:
3481:
3476:
3465:
3451:
3430:
3418:
3348:
3342:
3250:
3244:
3181:
3175:
3112:
3106:
3037:
3031:
2982:
2976:
2864:
2858:
2795:
2781:
2727:
2715:
2674:
2671:
2653:
2641:
2612:
2606:
2599:
2595:
2589:
2573:
2569:
2563:
2554:
2548:
2541:
2536:
2525:
2511:
2490:
2478:
2414:
2408:
2310:
2304:
2241:
2235:
2166:
2160:
2097:
2091:
2042:
2036:
1792:
1786:
1763:
1751:
1715:. The second case is similar.
1702:
1696:
1681:
1675:
1602:
1596:
1547:
1541:
1512:
1506:
1442:
1436:
1379:
1373:
1284:
1278:
1255:
1249:
1200:
1194:
1138:
1135:
1129:
1120:
1114:
1108:
1099:
1096:
1090:
1081:
1075:
1069:
1039:
1033:
995:
989:
966:
954:
925:
922:
916:
907:
901:
895:
880:
877:
871:
862:
856:
850:
827:
821:
798:
792:
737:
686:
674:
551:
545:
510:
504:
475:
469:
446:
434:
397:The intermediate value theorem
345:
339:
316:
310:
223:
217:
132:
126:
111:
105:
62:
50:
1:
10247:Belk, Jim (January 2, 2012).
10220:by Julio Cesar de la Yncera,
10073:Topology and Its Applications
10034:Topology and Its Applications
9801:Slightly modified version of
9747:Cauchy's Calcul Infinitésimal
9706:
7646:) is opposite to the sign of
5373:, and we have to prove, that
4135:{\displaystyle \forall x\in }
3436:{\displaystyle \forall x\in }
2496:{\displaystyle \forall x\in }
1297:, all points in the interval
388:
10233:"Intermediate Value Theorem"
10004:MVT: A Most Valuable Theorem
9979:"Intermediate value theorem"
9889:10.1016/0315-0860(80)90036-1
9479:Proof for 1-dimensional case
8997:{\displaystyle f(b)<f(a)}
8722:
8713:{\displaystyle f(a)<f(b)}
8599:
8354:{\displaystyle \mathbb {R} }
8136:In fact, connectedness is a
8129:
6472:{\displaystyle \delta >0}
5912:{\displaystyle \delta >0}
3138:{\displaystyle \varepsilon }
2198:{\displaystyle \varepsilon }
714:{\displaystyle \mathbb {R} }
7:
10086:10.1016/j.topol.2019.107036
10046:10.1016/j.topol.2015.12.066
9849:10.1007/978-1-4613-0087-8_2
9787:Clarke, Douglas A. (1971).
9672:
9134:In constructive mathematics
9128:Brouwer fixed-point theorem
8720:. Then once more invoking
8643:
8415:is a real number such that
8301:Consider a closed interval
8142:
8027:
7519:intermediate value property
6809:{\displaystyle a<x<c}
6216:{\displaystyle c<x<b}
5629:{\displaystyle S\subseteq }
5598:is not empty. Moreover, as
5266:{\displaystyle f(x)=g(x)+u}
5216:{\displaystyle g(x)=f(x)-u}
1638:{\displaystyle {\sqrt {2}}}
1206:{\displaystyle c,d\in f(I)}
10:
10307:
10203:Intermediate value theorem
8953:{\displaystyle c\in (a,b)}
8544:{\displaystyle c\in (a,b)}
8361:and a continuous function
8288:Intermediate value theorem
7986:{\displaystyle E\subset X}
7568:-dimensional simplex with
7369:is a real-valued function
7066:{\displaystyle f,\varphi }
7027:
6370:{\displaystyle g(c)\geq 0}
4273:{\displaystyle c\in (a,b)}
3954:{\displaystyle c\in (a,b)}
3786:{\displaystyle x\in I_{2}}
3711:{\displaystyle x\in I_{2}}
2841:{\displaystyle x\in I_{1}}
2766:{\displaystyle x\in I_{1}}
1461:{\displaystyle f(x)=x^{2}}
1399:is naturally contained in
1337:are also function values,
972:{\displaystyle c\in (a,b)}
721:and a continuous function
653:
252:intermediate value theorem
10291:Theorems in real analysis
10159:10.23638/LMCS-16(3:5)2020
9958:10.1007/s10699-011-9223-1
9755:10.1007/978-3-030-11036-9
9745:Cates, Dennis M. (2019).
9599:{\displaystyle f(A)-f(B)}
9328:{\displaystyle 0<f(b)}
9293:{\displaystyle f(a)<0}
8804:{\displaystyle u\in f(I)}
8224:is a continuous map, and
7967:is a continuous map, and
6937:{\displaystyle g(c)>0}
6777:{\displaystyle g(x)>0}
6703:{\displaystyle g(x)>0}
6598:. We can rewrite this as
6405:{\displaystyle g(c)>0}
6184:{\displaystyle g(x)<0}
6069:{\displaystyle g(x)<0}
6041:, which is equivalent to
5845:{\displaystyle g(c)<0}
5687:{\displaystyle c=\sup(S)}
5545:{\displaystyle g(a)<0}
3821:{\displaystyle f(x)>u}
3303:that are upper bounds of
3118:{\displaystyle u<f(b)}
2876:{\displaystyle f(x)<u}
2323:greater than or equal to
2178:{\displaystyle f(a)<u}
1804:{\displaystyle f(x)<u}
1645:is an irrational number.
145:. Then there exists some
9989:University of St Andrews
9807:. Springer. p. 123.
9431:A similar result is the
9140:constructive mathematics
9045:be a continuous map. If
8192:are topological spaces,
7852:)>0 for at least one
7834:on the face opposite to
7806:on the face opposite to
7661:on the face opposite to
7539:Poincaré-Miranda theorem
7533:Multi-dimensional spaces
7321:{\displaystyle \varphi }
3718:satisfies the condition
3145:is the distance between
2773:satisfies the condition
2205:is the distance between
1936:{\displaystyle c=\sup S}
1648:
1407:Relation to completeness
1330:{\displaystyle {\bigl }}
1151:{\displaystyle {\bigl }}
9803:Abbot, Stephen (2015).
9789:Foundations of Analysis
9620:, and as a consequence
9223:{\displaystyle f:\to R}
9123:is the order topology.
7572:+1 vertices denoted by
7504:Conway base 13 function
7234:{\displaystyle \alpha }
7086:{\displaystyle \alpha }
6882:, which contradict the
5223:which is equivalent to
5027:, from which we deduce
3916:{\displaystyle c\neq b}
3890:{\displaystyle c\neq a}
1911:, by completeness, the
361:This has two important
10108:Rudin, Walter (1976).
9950:Foundations of Science
9805:Understanding Analysis
9658:
9600:
9556:
9536:
9516:
9496:
9469:
9449:
9427:Practical applications
9416:
9375:
9355:
9329:
9294:
9259:
9224:
9180:
9160:
9027:set equipped with the
8998:
8954:
8916:
8866:
8865:{\displaystyle c\in I}
8840:
8839:{\displaystyle f(c)=u}
8805:
8770:
8714:
8670:
8635:
8580:
8579:{\displaystyle f(c)=u}
8545:
8507:
8409:
8389:
8355:
8333:
8271:
8238:
8218:
8186:
8166:
8122:
8055:
8020:
7987:
7961:
7925:
7905:
7802:)<0 for all points
7717:Then there is a point
7322:
7299:
7255:
7254:{\displaystyle \beta }
7235:
7215:
7195:
7151:
7107:
7106:{\displaystyle \beta }
7087:
7067:
7008:
7007:{\displaystyle f(c)=u}
6973:
6972:{\displaystyle g(c)=0}
6938:
6903:
6876:
6875:{\displaystyle x<c}
6850:
6836:is an upper bound for
6830:
6810:
6778:
6743:
6704:
6669:
6592:
6523:
6473:
6447:
6406:
6371:
6336:
6309:
6308:{\displaystyle x>c}
6283:
6269:is an upper bound for
6263:
6243:
6242:{\displaystyle x\in S}
6217:
6185:
6150:
6109:
6070:
6035:
5963:
5913:
5887:
5846:
5811:
5810:{\displaystyle g(c)=0}
5776:
5720:
5688:
5650:
5630:
5592:
5572:
5571:{\displaystyle a\in S}
5546:
5511:
5440:
5402:
5401:{\displaystyle g(c)=0}
5367:
5317:
5267:
5217:
5164:
5114:
5056:
5055:{\displaystyle f(c)=u}
5021:
4995:
4942:
4867:
4847:
4827:
4791:
4737:
4665:
4645:
4594:
4544:
4473:
4418:
4342:
4306:
4274:
4233:
4136:
4095:
4059:
4039:
4019:
3990:
3989:{\displaystyle f(c)=u}
3955:
3923:, it must be the case
3917:
3891:
3862:
3842:
3822:
3787:
3760:. Therefore for every
3754:
3712:
3679:
3634:
3574:Consider the interval
3568:
3437:
3396:
3367:
3317:
3297:
3277:
3257:
3228:
3214:sufficiently close to
3208:
3188:
3159:
3139:
3119:
3084:
3070:sufficiently close to
3064:
3044:
3015:
2989:
2960:
2940:
2917:
2897:
2877:
2842:
2815:. Therefore for every
2809:
2767:
2734:
2694:
2634:Consider the interval
2628:
2497:
2456:
2427:
2377:
2357:
2337:
2317:
2288:
2274:sufficiently close to
2268:
2248:
2219:
2199:
2179:
2144:
2130:sufficiently close to
2124:
2104:
2075:
2049:
2020:
2000:
1977:
1957:
1937:
1905:
1885:
1865:
1845:
1825:
1805:
1770:
1732:
1709:
1639:
1615:
1614:{\displaystyle f(x)=2}
1580:
1560:
1559:{\displaystyle f(2)=4}
1525:
1524:{\displaystyle f(0)=0}
1490:
1462:
1389:
1331:
1291:
1262:
1233:
1232:{\displaystyle c<d}
1207:
1152:
1046:
1008:
1007:{\displaystyle f(c)=u}
973:
935:
834:
805:
776:
749:
715:
693:
644:
624:
604:
584:
558:
557:{\displaystyle y=f(x)}
523:
522:{\displaystyle f(2)=5}
488:
487:{\displaystyle f(1)=3}
459:with the known values
453:
420:
398:
352:
323:
268:
243:
236:
235:{\displaystyle f(x)=s}
201:
181:
160:
139:
89:
69:
37:
9944:Karin Usadi Katz and
9659:
9601:
9557:
9537:
9517:
9497:
9470:
9455:-sphere to Euclidean
9450:
9417:
9376:
9361:there exists a point
9356:
9330:
9295:
9260:
9225:
9181:
9161:
8999:
8955:
8917:
8867:
8841:
8806:
8771:
8715:
8671:
8636:
8581:
8546:
8508:
8410:
8390:
8356:
8334:
8272:
8239:
8219:
8187:
8167:
8123:
8056:
8021:
7988:
7962:
7926:
7906:
7334:Joseph-Louis Lagrange
7330:Augustin-Louis Cauchy
7323:
7300:
7256:
7236:
7216:
7196:
7152:
7108:
7088:
7068:
7022:non-standard analysis
7009:
6974:
6939:
6904:
6877:
6851:
6831:
6811:
6779:
6744:
6705:
6670:
6593:
6524:
6474:
6448:
6412:. We similarly chose
6407:
6372:
6337:
6310:
6284:
6264:
6244:
6218:
6186:
6151:
6110:
6071:
6036:
5964:
5914:
5888:
5847:
5812:
5777:
5721:
5689:
5651:
5631:
5593:
5573:
5547:
5512:
5441:
5439:{\displaystyle c\in }
5403:
5368:
5318:
5268:
5218:
5165:
5115:
5057:
5022:
4996:
4943:
4868:
4848:
4828:
4792:
4738:
4666:
4651:that is contained in
4646:
4595:
4545:
4474:
4419:
4343:
4307:
4305:{\displaystyle (a,b)}
4275:
4234:
4137:
4096:
4060:
4040:
4020:
3991:
3956:
3918:
3892:
3863:
3843:
3823:
3788:
3755:
3713:
3680:
3635:
3569:
3438:
3397:
3368:
3318:
3298:
3278:
3258:
3229:
3209:
3189:
3160:
3140:
3120:
3085:
3065:
3045:
3016:
2990:
2961:
2941:
2918:
2898:
2878:
2843:
2810:
2768:
2735:
2695:
2629:
2498:
2457:
2428:
2378:
2358:
2338:
2318:
2289:
2269:
2249:
2220:
2200:
2180:
2145:
2125:
2105:
2076:
2050:
2021:
2001:
1978:
1958:
1938:
1906:
1886:
1866:
1846:
1826:
1806:
1771:
1769:{\displaystyle x\in }
1733:
1710:
1640:
1616:
1581:
1561:
1526:
1491:
1463:
1390:
1332:
1292:
1263:
1234:
1208:
1153:
1047:
1009:
974:
936:
835:
806:
777:
750:
716:
694:
661:Consider an interval
645:
625:
605:
585:
559:
524:
489:
454:
421:
396:
353:
324:
269:
248:mathematical analysis
237:
202:
182:
161:
140:
90:
70:
38:
22:
10286:Theorems in calculus
9975:Robertson, Edmund F.
9876:Historia Mathematica
9785:Essentially follows
9648:
9566:
9546:
9526:
9506:
9486:
9459:
9439:
9385:
9365:
9339:
9304:
9269:
9237:
9190:
9186:be real numbers and
9170:
9150:
9091:, then there exists
8964:
8926:
8876:
8850:
8815:
8780:
8730:
8680:
8669:{\displaystyle f(I)}
8651:
8607:
8555:
8517:
8419:
8399:
8365:
8343:
8339:in the real numbers
8305:
8270:{\displaystyle f(X)}
8252:
8228:
8196:
8176:
8156:
8138:topological property
8065:
8037:
8019:{\displaystyle f(E)}
8001:
7971:
7939:
7915:
7895:
7312:
7265:
7245:
7225:
7205:
7161:
7117:
7097:
7077:
7051:
6983:
6948:
6913:
6909:, which means, that
6893:
6860:
6840:
6820:
6788:
6753:
6714:
6679:
6675:which implies, that
6602:
6533:
6483:
6457:
6416:
6381:
6377:. Assume then, that
6346:
6326:
6315:, contradicting the
6293:
6273:
6253:
6227:
6195:
6160:
6119:
6080:
6045:
5973:
5923:
5897:
5856:
5821:
5786:
5730:
5719:{\displaystyle g(c)}
5701:
5663:
5640:
5602:
5582:
5556:
5521:
5450:
5412:
5377:
5327:
5277:
5273:and lets us rewrite
5227:
5177:
5124:
5074:
5031:
5005:
4952:
4877:
4857:
4837:
4801:
4747:
4675:
4655:
4604:
4554:
4483:
4428:
4352:
4316:
4284:
4246:
4146:
4105:
4069:
4049:
4029:
4003:
3965:
3961:. Now we claim that
3927:
3901:
3875:
3852:
3832:
3797:
3764:
3722:
3689:
3644:
3578:
3447:
3406:
3377:
3330:
3307:
3287:
3267:
3256:{\displaystyle f(x)}
3238:
3218:
3198:
3187:{\displaystyle f(b)}
3169:
3149:
3129:
3094:
3074:
3054:
3043:{\displaystyle f(b)}
3025:
2999:
2988:{\displaystyle f(x)}
2970:
2950:
2930:
2907:
2887:
2852:
2819:
2777:
2744:
2699:
2638:
2507:
2466:
2437:
2390:
2367:
2347:
2327:
2316:{\displaystyle f(x)}
2298:
2278:
2258:
2247:{\displaystyle f(a)}
2229:
2209:
2189:
2154:
2134:
2114:
2103:{\displaystyle f(a)}
2085:
2059:
2048:{\displaystyle f(x)}
2030:
2010:
1990:
1967:
1947:
1918:
1895:
1875:
1855:
1835:
1815:
1780:
1742:
1722:
1669:
1625:
1590:
1570:
1535:
1500:
1472:
1430:
1341:
1301:
1290:{\displaystyle f(b)}
1272:
1261:{\displaystyle f(a)}
1243:
1217:
1176:
1056:
1045:{\displaystyle f(I)}
1027:
983:
945:
844:
833:{\displaystyle f(b)}
815:
804:{\displaystyle f(a)}
786:
782:is a number between
766:
725:
703:
665:
634:
614:
594:
568:
533:
529:, then the graph of
498:
463:
431:
410:
351:{\displaystyle f(b)}
333:
322:{\displaystyle f(a)}
304:
258:
211:
191:
171:
150:
99:
79:
47:
27:
9973:O'Connor, John J.;
9870:Russ, S.B. (1980).
9725:"Bolzano's Theorem"
9433:Borsuk–Ulam theorem
8299: —
7671:The sign-vector of
7201:. Then there is an
7038:squaring the circle
6076:. If we just chose
4853:is the supremum of
1831:is non-empty since
583:{\displaystyle y=4}
10230:Weisstein, Eric W.
9722:Weisstein, Eric W.
9694:Hairy ball theorem
9685:Non-atomic measure
9679:Mean value theorem
9654:
9596:
9552:
9532:
9512:
9492:
9480:
9465:
9445:
9412:
9371:
9351:
9325:
9290:
9255:
9220:
9176:
9156:
9053:are two points in
9004:, so we are done.
8994:
8950:
8912:
8862:
8836:
8801:
8766:
8710:
8666:
8634:{\displaystyle I=}
8631:
8595:
8576:
8541:
8503:
8405:
8385:
8351:
8332:{\displaystyle I=}
8329:
8289:
8267:
8234:
8214:
8182:
8162:
8148:topological spaces
8118:
8051:
8016:
7983:
7957:
7921:
7901:
7318:
7295:
7251:
7231:
7211:
7191:
7147:
7103:
7083:
7063:
7034:Bryson of Heraclea
7004:
6969:
6934:
6899:
6872:
6846:
6826:
6816:. It follows that
6806:
6774:
6739:
6710:. If we now chose
6700:
6665:
6588:
6519:
6469:
6443:
6402:
6367:
6332:
6305:
6279:
6259:
6249:. It follows that
6239:
6213:
6181:
6146:
6105:
6066:
6031:
5959:
5909:
5883:
5842:
5807:
5772:
5716:
5684:
5646:
5626:
5588:
5568:
5542:
5507:
5436:
5398:
5363:
5313:
5263:
5213:
5160:
5110:
5052:
5017:
5001:are valid for all
4991:
4948:Both inequalities
4938:
4873:. This means that
4863:
4843:
4823:
4787:
4733:
4661:
4641:
4590:
4540:
4469:
4414:
4338:
4302:
4270:
4229:
4132:
4091:
4055:
4035:
4015:
3986:
3951:
3913:
3887:
3858:
3838:
3818:
3783:
3750:
3708:
3675:
3630:
3564:
3433:
3392:
3363:
3313:
3293:
3273:
3253:
3224:
3204:
3184:
3155:
3135:
3115:
3080:
3060:
3040:
3011:
2985:
2956:
2936:
2913:
2893:
2873:
2838:
2805:
2763:
2730:
2690:
2624:
2493:
2452:
2423:
2373:
2353:
2333:
2313:
2284:
2264:
2244:
2215:
2195:
2175:
2140:
2120:
2100:
2071:
2045:
2016:
1996:
1973:
1953:
1933:
1901:
1881:
1861:
1841:
1821:
1801:
1766:
1738:be the set of all
1728:
1705:
1635:
1611:
1576:
1556:
1521:
1486:
1458:
1424:irrational numbers
1385:
1327:
1287:
1258:
1229:
1203:
1148:
1042:
1004:
969:
931:
830:
801:
772:
745:
711:
692:{\displaystyle I=}
689:
640:
620:
600:
580:
554:
519:
484:
449:
416:
399:
373:in that interval (
348:
319:
264:
244:
232:
197:
177:
156:
135:
85:
65:
33:
10218:Bolzano's Theorem
10121:978-0-07-054235-8
9764:978-3-030-11035-2
9657:{\displaystyle n}
9555:{\displaystyle d}
9535:{\displaystyle B}
9515:{\displaystyle A}
9495:{\displaystyle f}
9478:
9468:{\displaystyle n}
9448:{\displaystyle n}
9374:{\displaystyle x}
9179:{\displaystyle b}
9159:{\displaystyle a}
8593:
8408:{\displaystyle u}
8287:
8237:{\displaystyle X}
8185:{\displaystyle Y}
8165:{\displaystyle X}
8088:
7924:{\displaystyle Y}
7904:{\displaystyle X}
7813:. In particular,
7657:) for all points
7511:Darboux's theorem
7381:in the domain of
7361:Converse is false
7214:{\displaystyle x}
6902:{\displaystyle c}
6888:least upper bound
6849:{\displaystyle S}
6829:{\displaystyle x}
6737:
6335:{\displaystyle c}
6321:least upper bound
6282:{\displaystyle S}
6262:{\displaystyle x}
6144:
6103:
5893:, there exists a
5649:{\displaystyle S}
5591:{\displaystyle S}
5170:case is similar.
4866:{\displaystyle S}
4846:{\displaystyle c}
4664:{\displaystyle S}
4058:{\displaystyle c}
4045:is continuous at
4038:{\displaystyle f}
3861:{\displaystyle b}
3841:{\displaystyle c}
3316:{\displaystyle S}
3296:{\displaystyle b}
3276:{\displaystyle u}
3227:{\displaystyle b}
3207:{\displaystyle x}
3158:{\displaystyle u}
3083:{\displaystyle b}
3063:{\displaystyle x}
2959:{\displaystyle b}
2939:{\displaystyle f}
2916:{\displaystyle a}
2896:{\displaystyle c}
2376:{\displaystyle S}
2356:{\displaystyle a}
2336:{\displaystyle u}
2287:{\displaystyle a}
2267:{\displaystyle x}
2218:{\displaystyle u}
2143:{\displaystyle a}
2123:{\displaystyle x}
2019:{\displaystyle a}
1999:{\displaystyle f}
1976:{\displaystyle S}
1956:{\displaystyle c}
1943:exists. That is,
1904:{\displaystyle b}
1884:{\displaystyle S}
1864:{\displaystyle S}
1851:is an element of
1844:{\displaystyle a}
1824:{\displaystyle S}
1731:{\displaystyle S}
1633:
1579:{\displaystyle x}
775:{\displaystyle u}
643:{\displaystyle 2}
623:{\displaystyle 1}
603:{\displaystyle x}
419:{\displaystyle f}
375:Bolzano's theorem
267:{\displaystyle f}
200:{\displaystyle b}
180:{\displaystyle a}
159:{\displaystyle x}
95:be a number with
88:{\displaystyle s}
36:{\displaystyle f}
10298:
10258:
10243:
10242:
10178:
10169:
10163:
10162:
10152:
10132:
10126:
10125:
10105:
10099:
10098:
10088:
10064:
10058:
10057:
10025:
10019:
10018:
9998:
9992:
9991:
9970:
9964:
9942:
9936:
9935:
9909:
9900:
9894:
9893:
9891:
9867:
9861:
9860:
9836:
9830:
9829:
9827:
9815:
9809:
9808:
9799:
9793:
9792:
9783:
9777:
9776:
9742:
9736:
9735:
9734:
9717:
9690:
9665:
9663:
9661:
9660:
9655:
9638:
9619:
9612:
9605:
9603:
9602:
9597:
9561:
9559:
9558:
9553:
9541:
9539:
9538:
9533:
9521:
9519:
9518:
9513:
9501:
9499:
9498:
9493:
9474:
9472:
9471:
9466:
9454:
9452:
9451:
9446:
9421:
9419:
9418:
9413:
9380:
9378:
9377:
9372:
9360:
9358:
9357:
9352:
9334:
9332:
9331:
9326:
9299:
9297:
9296:
9291:
9264:
9262:
9261:
9258:{\displaystyle }
9256:
9229:
9227:
9226:
9221:
9185:
9183:
9182:
9177:
9165:
9163:
9162:
9157:
9118:
9112:
9098:
9094:
9090:
9087:with respect to
9086:
9075:
9064:
9060:
9056:
9052:
9048:
9044:
9022:
9014:
9003:
9001:
9000:
8995:
8959:
8957:
8956:
8951:
8921:
8919:
8918:
8913:
8871:
8869:
8868:
8863:
8845:
8843:
8842:
8837:
8810:
8808:
8807:
8802:
8775:
8773:
8772:
8767:
8719:
8717:
8716:
8711:
8675:
8673:
8672:
8667:
8647:that the image,
8640:
8638:
8637:
8632:
8585:
8583:
8582:
8577:
8550:
8548:
8547:
8542:
8512:
8510:
8509:
8504:
8414:
8412:
8411:
8406:
8394:
8392:
8391:
8386:
8384:
8360:
8358:
8357:
8352:
8350:
8338:
8336:
8335:
8330:
8300:
8297:
8276:
8274:
8273:
8268:
8243:
8241:
8240:
8235:
8223:
8221:
8220:
8215:
8191:
8189:
8188:
8183:
8171:
8169:
8168:
8163:
8131:
8127:
8125:
8124:
8119:
8086:
8060:
8058:
8057:
8052:
8050:
8029:
8025:
8023:
8022:
8017:
7992:
7990:
7989:
7984:
7966:
7964:
7963:
7958:
7930:
7928:
7927:
7922:
7910:
7908:
7907:
7902:
7501:
7497:
7482:
7475:
7468:
7461:
7446:
7436:
7422:
7418:
7414:
7411:, there is some
7410:
7399:
7388:
7384:
7380:
7376:
7372:
7367:Darboux function
7327:
7325:
7324:
7319:
7304:
7302:
7301:
7296:
7260:
7258:
7257:
7252:
7240:
7238:
7237:
7232:
7220:
7218:
7217:
7212:
7200:
7198:
7197:
7192:
7156:
7154:
7153:
7148:
7112:
7110:
7109:
7104:
7092:
7090:
7089:
7084:
7072:
7070:
7069:
7064:
7013:
7011:
7010:
7005:
6978:
6976:
6975:
6970:
6943:
6941:
6940:
6935:
6908:
6906:
6905:
6900:
6886:property of the
6881:
6879:
6878:
6873:
6855:
6853:
6852:
6847:
6835:
6833:
6832:
6827:
6815:
6813:
6812:
6807:
6783:
6781:
6780:
6775:
6748:
6746:
6745:
6740:
6738:
6730:
6709:
6707:
6706:
6701:
6674:
6672:
6671:
6666:
6597:
6595:
6594:
6589:
6572:
6540:
6528:
6526:
6525:
6520:
6478:
6476:
6475:
6470:
6452:
6450:
6449:
6444:
6411:
6409:
6408:
6403:
6376:
6374:
6373:
6368:
6341:
6339:
6338:
6333:
6319:property of the
6314:
6312:
6311:
6306:
6288:
6286:
6285:
6280:
6268:
6266:
6265:
6260:
6248:
6246:
6245:
6240:
6222:
6220:
6219:
6214:
6190:
6188:
6187:
6182:
6155:
6153:
6152:
6147:
6145:
6143:
6129:
6114:
6112:
6111:
6106:
6104:
6096:
6075:
6073:
6072:
6067:
6040:
6038:
6037:
6032:
6012:
5980:
5968:
5966:
5965:
5960:
5918:
5916:
5915:
5910:
5892:
5890:
5889:
5884:
5851:
5849:
5848:
5843:
5816:
5814:
5813:
5808:
5781:
5779:
5778:
5773:
5725:
5723:
5722:
5717:
5693:
5691:
5690:
5685:
5655:
5653:
5652:
5647:
5635:
5633:
5632:
5627:
5597:
5595:
5594:
5589:
5577:
5575:
5574:
5569:
5551:
5549:
5548:
5543:
5516:
5514:
5513:
5508:
5445:
5443:
5442:
5437:
5407:
5405:
5404:
5399:
5372:
5370:
5369:
5364:
5322:
5320:
5319:
5314:
5272:
5270:
5269:
5264:
5222:
5220:
5219:
5214:
5169:
5167:
5166:
5161:
5119:
5117:
5116:
5111:
5061:
5059:
5058:
5053:
5026:
5024:
5023:
5018:
5000:
4998:
4997:
4992:
4947:
4945:
4944:
4939:
4913:
4912:
4872:
4870:
4869:
4864:
4852:
4850:
4849:
4844:
4832:
4830:
4829:
4824:
4816:
4815:
4796:
4794:
4793:
4788:
4762:
4761:
4742:
4740:
4739:
4734:
4708:
4707:
4670:
4668:
4667:
4662:
4650:
4648:
4647:
4642:
4616:
4615:
4599:
4597:
4596:
4591:
4549:
4547:
4546:
4541:
4478:
4476:
4475:
4470:
4465:
4464:
4452:
4451:
4423:
4421:
4420:
4415:
4392:
4391:
4373:
4372:
4347:
4345:
4344:
4339:
4331:
4330:
4311:
4309:
4308:
4303:
4279:
4277:
4276:
4271:
4238:
4236:
4235:
4230:
4222:
4190:
4180:
4179:
4167:
4153:
4141:
4139:
4138:
4133:
4100:
4098:
4097:
4092:
4084:
4083:
4064:
4062:
4061:
4056:
4044:
4042:
4041:
4036:
4024:
4022:
4021:
4016:
3995:
3993:
3992:
3987:
3960:
3958:
3957:
3952:
3922:
3920:
3919:
3914:
3896:
3894:
3893:
3888:
3867:
3865:
3864:
3859:
3847:
3845:
3844:
3839:
3827:
3825:
3824:
3819:
3792:
3790:
3789:
3784:
3782:
3781:
3759:
3757:
3756:
3751:
3743:
3729:
3717:
3715:
3714:
3709:
3707:
3706:
3684:
3682:
3681:
3676:
3656:
3655:
3639:
3637:
3636:
3631:
3629:
3628:
3573:
3571:
3570:
3565:
3516:
3484:
3468:
3454:
3442:
3440:
3439:
3434:
3401:
3399:
3398:
3393:
3372:
3370:
3369:
3364:
3322:
3320:
3319:
3314:
3302:
3300:
3299:
3294:
3282:
3280:
3279:
3274:
3262:
3260:
3259:
3254:
3233:
3231:
3230:
3225:
3213:
3211:
3210:
3205:
3193:
3191:
3190:
3185:
3164:
3162:
3161:
3156:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3089:
3087:
3086:
3081:
3069:
3067:
3066:
3061:
3049:
3047:
3046:
3041:
3020:
3018:
3017:
3012:
2994:
2992:
2991:
2986:
2965:
2963:
2962:
2957:
2945:
2943:
2942:
2937:
2922:
2920:
2919:
2914:
2902:
2900:
2899:
2894:
2882:
2880:
2879:
2874:
2847:
2845:
2844:
2839:
2837:
2836:
2814:
2812:
2811:
2806:
2798:
2784:
2772:
2770:
2769:
2764:
2762:
2761:
2739:
2737:
2736:
2731:
2711:
2710:
2697:
2696:
2691:
2689:
2688:
2633:
2631:
2630:
2625:
2576:
2544:
2528:
2514:
2502:
2500:
2499:
2494:
2461:
2459:
2458:
2453:
2432:
2430:
2429:
2424:
2382:
2380:
2379:
2374:
2362:
2360:
2359:
2354:
2342:
2340:
2339:
2334:
2322:
2320:
2319:
2314:
2293:
2291:
2290:
2285:
2273:
2271:
2270:
2265:
2253:
2251:
2250:
2245:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2184:
2182:
2181:
2176:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2109:
2107:
2106:
2101:
2080:
2078:
2077:
2072:
2054:
2052:
2051:
2046:
2025:
2023:
2022:
2017:
2005:
2003:
2002:
1997:
1982:
1980:
1979:
1974:
1962:
1960:
1959:
1954:
1942:
1940:
1939:
1934:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1830:
1828:
1827:
1822:
1810:
1808:
1807:
1802:
1775:
1773:
1772:
1767:
1737:
1735:
1734:
1729:
1714:
1712:
1711:
1706:
1644:
1642:
1641:
1636:
1634:
1629:
1620:
1618:
1617:
1612:
1585:
1583:
1582:
1577:
1565:
1563:
1562:
1557:
1530:
1528:
1527:
1522:
1495:
1493:
1492:
1487:
1485:
1467:
1465:
1464:
1459:
1457:
1456:
1417:rational numbers
1394:
1392:
1391:
1386:
1366:
1365:
1350:
1349:
1336:
1334:
1333:
1328:
1326:
1325:
1310:
1309:
1296:
1294:
1293:
1288:
1267:
1265:
1264:
1259:
1238:
1236:
1235:
1230:
1212:
1210:
1209:
1204:
1168:states that the
1157:
1155:
1154:
1149:
1147:
1146:
1065:
1064:
1051:
1049:
1048:
1043:
1013:
1011:
1010:
1005:
978:
976:
975:
970:
941:then there is a
940:
938:
937:
932:
839:
837:
836:
831:
810:
808:
807:
802:
781:
779:
778:
773:
754:
752:
751:
746:
744:
720:
718:
717:
712:
710:
699:of real numbers
698:
696:
695:
690:
649:
647:
646:
641:
629:
627:
626:
621:
609:
607:
606:
601:
589:
587:
586:
581:
563:
561:
560:
555:
528:
526:
525:
520:
493:
491:
490:
485:
458:
456:
455:
452:{\displaystyle }
450:
425:
423:
422:
417:
357:
355:
354:
349:
328:
326:
325:
320:
299:
273:
271:
270:
265:
241:
239:
238:
233:
206:
204:
203:
198:
186:
184:
183:
178:
165:
163:
162:
157:
144:
142:
141:
136:
94:
92:
91:
86:
74:
72:
71:
68:{\displaystyle }
66:
42:
40:
39:
34:
10306:
10305:
10301:
10300:
10299:
10297:
10296:
10295:
10271:
10270:
10199:
10187:
10185:Further reading
10182:
10181:
10170:
10166:
10133:
10129:
10122:
10106:
10102:
10065:
10061:
10026:
10022:
10015:
9999:
9995:
9971:
9967:
9946:Mikhail G. Katz
9943:
9939:
9924:10.2307/2975545
9907:
9901:
9897:
9868:
9864:
9837:
9833:
9816:
9812:
9800:
9796:
9784:
9780:
9765:
9749:. p. 249.
9743:
9739:
9718:
9714:
9709:
9700:Sperner's lemma
9688:
9675:
9649:
9646:
9645:
9644:
9641:
9639:at this angle.
9621:
9614:
9607:
9567:
9564:
9563:
9547:
9544:
9543:
9527:
9524:
9523:
9507:
9504:
9503:
9487:
9484:
9483:
9460:
9457:
9456:
9440:
9437:
9436:
9429:
9386:
9383:
9382:
9366:
9363:
9362:
9340:
9337:
9336:
9305:
9302:
9301:
9270:
9267:
9266:
9238:
9235:
9234:
9232:closed interval
9191:
9188:
9187:
9171:
9168:
9167:
9151:
9148:
9147:
9136:
9114:
9100:
9096:
9092:
9088:
9077:
9066:
9062:
9058:
9054:
9050:
9046:
9032:
9025:totally ordered
9016:
9012:
9009:
8965:
8962:
8961:
8927:
8924:
8923:
8877:
8874:
8873:
8851:
8848:
8847:
8816:
8813:
8812:
8781:
8778:
8777:
8731:
8728:
8727:
8681:
8678:
8677:
8652:
8649:
8648:
8608:
8605:
8604:
8588:
8556:
8553:
8552:
8518:
8515:
8514:
8513:, there exists
8420:
8417:
8416:
8400:
8397:
8396:
8380:
8366:
8363:
8362:
8346:
8344:
8341:
8340:
8306:
8303:
8302:
8298:
8291:
8253:
8250:
8249:
8246:connected space
8229:
8226:
8225:
8197:
8194:
8193:
8177:
8174:
8173:
8157:
8154:
8153:
8146:generalizes to
8066:
8063:
8062:
8046:
8038:
8035:
8034:
8026:is connected. (
8002:
7999:
7998:
7972:
7969:
7968:
7940:
7937:
7936:
7916:
7913:
7912:
7896:
7893:
7892:
7887:in particular:
7873:
7846:
7839:
7830:For all points
7825:
7818:
7811:
7796:
7789:
7782:
7761:)>0 for all
7759:
7752:
7745:
7711:
7704:
7698:
7691:
7683:
7677:
7666:
7651:
7644:
7637:
7602:
7596:
7584:
7578:
7535:
7530:
7528:Generalizations
7515:differentiation
7499:
7488:
7477:
7470:
7463:
7448:
7441:
7424:
7420:
7416:
7412:
7401:
7390:
7386:
7382:
7378:
7374:
7370:
7363:
7313:
7310:
7309:
7266:
7263:
7262:
7246:
7243:
7242:
7226:
7223:
7222:
7206:
7203:
7202:
7162:
7159:
7158:
7118:
7115:
7114:
7098:
7095:
7094:
7078:
7075:
7074:
7052:
7049:
7048:
7042:Bernard Bolzano
7030:
6984:
6981:
6980:
6949:
6946:
6945:
6914:
6911:
6910:
6894:
6891:
6890:
6861:
6858:
6857:
6841:
6838:
6837:
6821:
6818:
6817:
6789:
6786:
6785:
6754:
6751:
6750:
6729:
6715:
6712:
6711:
6680:
6677:
6676:
6603:
6600:
6599:
6568:
6536:
6534:
6531:
6530:
6484:
6481:
6480:
6458:
6455:
6454:
6417:
6414:
6413:
6382:
6379:
6378:
6347:
6344:
6343:
6327:
6324:
6323:
6294:
6291:
6290:
6274:
6271:
6270:
6254:
6251:
6250:
6228:
6225:
6224:
6196:
6193:
6192:
6161:
6158:
6157:
6133:
6128:
6120:
6117:
6116:
6095:
6081:
6078:
6077:
6046:
6043:
6042:
6008:
5976:
5974:
5971:
5970:
5924:
5921:
5920:
5898:
5895:
5894:
5857:
5854:
5853:
5822:
5819:
5818:
5787:
5784:
5783:
5731:
5728:
5727:
5702:
5699:
5698:
5664:
5661:
5660:
5641:
5638:
5637:
5636:, we know that
5603:
5600:
5599:
5583:
5580:
5579:
5557:
5554:
5553:
5522:
5519:
5518:
5451:
5448:
5447:
5413:
5410:
5409:
5378:
5375:
5374:
5328:
5325:
5324:
5278:
5275:
5274:
5228:
5225:
5224:
5178:
5175:
5174:
5125:
5122:
5121:
5075:
5072:
5071:
5068:
5066:Proof version B
5032:
5029:
5028:
5006:
5003:
5002:
4953:
4950:
4949:
4905:
4901:
4878:
4875:
4874:
4858:
4855:
4854:
4838:
4835:
4834:
4808:
4804:
4802:
4799:
4798:
4797:, we know that
4754:
4750:
4748:
4745:
4744:
4703:
4699:
4676:
4673:
4672:
4656:
4653:
4652:
4611:
4607:
4605:
4602:
4601:
4555:
4552:
4551:
4484:
4481:
4480:
4479:. Then we have
4460:
4456:
4447:
4443:
4429:
4426:
4425:
4387:
4383:
4368:
4364:
4353:
4350:
4349:
4326:
4322:
4317:
4314:
4313:
4285:
4282:
4281:
4247:
4244:
4243:
4218:
4186:
4175:
4171:
4163:
4149:
4147:
4144:
4143:
4106:
4103:
4102:
4079:
4075:
4070:
4067:
4066:
4050:
4047:
4046:
4030:
4027:
4026:
4004:
4001:
4000:
3966:
3963:
3962:
3928:
3925:
3924:
3902:
3899:
3898:
3876:
3873:
3872:
3853:
3850:
3849:
3833:
3830:
3829:
3798:
3795:
3794:
3777:
3773:
3765:
3762:
3761:
3739:
3725:
3723:
3720:
3719:
3702:
3698:
3690:
3687:
3686:
3651:
3647:
3645:
3642:
3641:
3624:
3620:
3579:
3576:
3575:
3512:
3480:
3464:
3450:
3448:
3445:
3444:
3407:
3404:
3403:
3378:
3375:
3374:
3331:
3328:
3327:
3308:
3305:
3304:
3288:
3285:
3284:
3268:
3265:
3264:
3239:
3236:
3235:
3234:must then make
3219:
3216:
3215:
3199:
3196:
3195:
3170:
3167:
3166:
3150:
3147:
3146:
3130:
3127:
3126:
3095:
3092:
3091:
3075:
3072:
3071:
3055:
3052:
3051:
3026:
3023:
3022:
3000:
2997:
2996:
2971:
2968:
2967:
2951:
2948:
2947:
2931:
2928:
2927:
2908:
2905:
2904:
2888:
2885:
2884:
2853:
2850:
2849:
2832:
2828:
2820:
2817:
2816:
2794:
2780:
2778:
2775:
2774:
2757:
2753:
2745:
2742:
2741:
2706:
2702:
2700:
2684:
2680:
2639:
2636:
2635:
2572:
2540:
2524:
2510:
2508:
2505:
2504:
2467:
2464:
2463:
2438:
2435:
2434:
2391:
2388:
2387:
2368:
2365:
2364:
2348:
2345:
2344:
2328:
2325:
2324:
2299:
2296:
2295:
2279:
2276:
2275:
2259:
2256:
2255:
2230:
2227:
2226:
2210:
2207:
2206:
2190:
2187:
2186:
2155:
2152:
2151:
2135:
2132:
2131:
2115:
2112:
2111:
2086:
2083:
2082:
2060:
2057:
2056:
2031:
2028:
2027:
2011:
2008:
2007:
1991:
1988:
1987:
1968:
1965:
1964:
1948:
1945:
1944:
1919:
1916:
1915:
1896:
1893:
1892:
1876:
1873:
1872:
1856:
1853:
1852:
1836:
1833:
1832:
1816:
1813:
1812:
1781:
1778:
1777:
1743:
1740:
1739:
1723:
1720:
1719:
1670:
1667:
1666:
1656:
1654:Proof version A
1651:
1628:
1626:
1623:
1622:
1591:
1588:
1587:
1571:
1568:
1567:
1536:
1533:
1532:
1501:
1498:
1497:
1481:
1473:
1470:
1469:
1452:
1448:
1431:
1428:
1427:
1409:
1361:
1360:
1345:
1344:
1342:
1339:
1338:
1321:
1320:
1305:
1304:
1302:
1299:
1298:
1273:
1270:
1269:
1244:
1241:
1240:
1218:
1215:
1214:
1177:
1174:
1173:
1142:
1141:
1060:
1059:
1057:
1054:
1053:
1028:
1025:
1024:
984:
981:
980:
946:
943:
942:
845:
842:
841:
816:
813:
812:
787:
784:
783:
767:
764:
763:
740:
726:
723:
722:
706:
704:
701:
700:
666:
663:
662:
656:
635:
632:
631:
615:
612:
611:
595:
592:
591:
569:
566:
565:
534:
531:
530:
499:
496:
495:
464:
461:
460:
432:
429:
428:
411:
408:
407:
391:
334:
331:
330:
305:
302:
301:
289:
259:
256:
255:
254:states that if
212:
209:
208:
192:
189:
188:
172:
169:
168:
151:
148:
147:
100:
97:
96:
80:
77:
76:
48:
45:
44:
28:
25:
24:
17:
12:
11:
5:
10304:
10294:
10293:
10288:
10283:
10269:
10268:
10259:
10254:Stack Exchange
10244:
10225:
10215:
10206:
10198:
10197:External links
10195:
10194:
10193:
10186:
10183:
10180:
10179:
10164:
10127:
10120:
10100:
10059:
10020:
10013:
9993:
9965:
9937:
9918:(3): 185–194.
9895:
9882:(2): 156–185.
9862:
9831:
9810:
9794:
9778:
9763:
9737:
9711:
9710:
9708:
9705:
9704:
9703:
9697:
9691:
9682:
9674:
9671:
9653:
9595:
9592:
9589:
9586:
9583:
9580:
9577:
9574:
9571:
9551:
9531:
9511:
9491:
9477:
9464:
9444:
9428:
9425:
9424:
9423:
9411:
9408:
9405:
9402:
9399:
9396:
9393:
9390:
9370:
9350:
9347:
9344:
9324:
9321:
9318:
9315:
9312:
9309:
9289:
9286:
9283:
9280:
9277:
9274:
9254:
9251:
9248:
9245:
9242:
9219:
9216:
9213:
9210:
9207:
9204:
9201:
9198:
9195:
9175:
9155:
9135:
9132:
9065:lying between
9061:is a point in
9029:order topology
8993:
8990:
8987:
8984:
8981:
8978:
8975:
8972:
8969:
8949:
8946:
8943:
8940:
8937:
8934:
8931:
8911:
8908:
8905:
8902:
8899:
8896:
8893:
8890:
8887:
8884:
8881:
8861:
8858:
8855:
8835:
8832:
8829:
8826:
8823:
8820:
8800:
8797:
8794:
8791:
8788:
8785:
8765:
8762:
8759:
8756:
8753:
8750:
8747:
8744:
8741:
8738:
8735:
8709:
8706:
8703:
8700:
8697:
8694:
8691:
8688:
8685:
8665:
8662:
8659:
8656:
8630:
8627:
8624:
8621:
8618:
8615:
8612:
8592:
8575:
8572:
8569:
8566:
8563:
8560:
8540:
8537:
8534:
8531:
8528:
8525:
8522:
8502:
8499:
8496:
8493:
8490:
8487:
8484:
8481:
8478:
8475:
8472:
8469:
8466:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8442:
8439:
8436:
8433:
8430:
8427:
8424:
8404:
8383:
8379:
8376:
8373:
8370:
8349:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8285:
8266:
8263:
8260:
8257:
8233:
8213:
8210:
8207:
8204:
8201:
8181:
8161:
8134:
8133:
8117:
8114:
8111:
8107:
8103:
8100:
8097:
8094:
8091:
8085:
8082:
8079:
8076:
8073:
8070:
8049:
8045:
8042:
8031:
8015:
8012:
8009:
8006:
7982:
7979:
7976:
7956:
7953:
7950:
7947:
7944:
7920:
7900:
7872:
7869:
7861:
7860:
7844:
7837:
7828:
7823:
7816:
7809:
7794:
7787:
7780:
7757:
7750:
7743:
7715:
7714:
7709:
7702:
7696:
7689:
7681:
7675:
7669:
7664:
7649:
7642:
7635:
7632:, the sign of
7600:
7594:
7582:
7576:
7534:
7531:
7529:
7526:
7362:
7359:
7355:infinitesimals
7351:Louis Arbogast
7317:
7294:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7250:
7230:
7210:
7190:
7187:
7184:
7181:
7178:
7175:
7172:
7169:
7166:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7125:
7122:
7102:
7082:
7062:
7059:
7056:
7029:
7026:
7003:
7000:
6997:
6994:
6991:
6988:
6968:
6965:
6962:
6959:
6956:
6953:
6933:
6930:
6927:
6924:
6921:
6918:
6898:
6871:
6868:
6865:
6845:
6825:
6805:
6802:
6799:
6796:
6793:
6773:
6770:
6767:
6764:
6761:
6758:
6736:
6733:
6728:
6725:
6722:
6719:
6699:
6696:
6693:
6690:
6687:
6684:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6607:
6587:
6584:
6581:
6578:
6575:
6571:
6567:
6564:
6561:
6558:
6555:
6552:
6549:
6546:
6543:
6539:
6518:
6515:
6512:
6509:
6506:
6503:
6500:
6497:
6494:
6491:
6488:
6468:
6465:
6462:
6442:
6439:
6436:
6433:
6430:
6427:
6424:
6421:
6401:
6398:
6395:
6392:
6389:
6386:
6366:
6363:
6360:
6357:
6354:
6351:
6331:
6304:
6301:
6298:
6278:
6258:
6238:
6235:
6232:
6212:
6209:
6206:
6203:
6200:
6180:
6177:
6174:
6171:
6168:
6165:
6142:
6139:
6136:
6132:
6127:
6124:
6102:
6099:
6094:
6091:
6088:
6085:
6065:
6062:
6059:
6056:
6053:
6050:
6030:
6027:
6024:
6021:
6018:
6015:
6011:
6007:
6004:
6001:
5998:
5995:
5992:
5989:
5986:
5983:
5979:
5969:implies, that
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5908:
5905:
5902:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5841:
5838:
5835:
5832:
5829:
5826:
5806:
5803:
5800:
5797:
5794:
5791:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5741:
5738:
5735:
5726:, those being
5715:
5712:
5709:
5706:
5683:
5680:
5677:
5674:
5671:
5668:
5645:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5587:
5567:
5564:
5561:
5552:we know, that
5541:
5538:
5535:
5532:
5529:
5526:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5397:
5394:
5391:
5388:
5385:
5382:
5362:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5312:
5309:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5212:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5159:
5156:
5153:
5150:
5147:
5144:
5141:
5138:
5135:
5132:
5129:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5079:
5067:
5064:
5051:
5048:
5045:
5042:
5039:
5036:
5016:
5013:
5010:
4990:
4987:
4984:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4911:
4908:
4904:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4862:
4842:
4822:
4819:
4814:
4811:
4807:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4760:
4757:
4753:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4706:
4702:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4660:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4614:
4610:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4468:
4463:
4459:
4455:
4450:
4446:
4442:
4439:
4436:
4433:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4390:
4386:
4382:
4379:
4376:
4371:
4367:
4363:
4360:
4357:
4337:
4334:
4329:
4325:
4321:
4301:
4298:
4295:
4292:
4289:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4228:
4225:
4221:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4189:
4184:
4178:
4174:
4170:
4166:
4162:
4159:
4156:
4152:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4090:
4087:
4082:
4078:
4074:
4054:
4034:
4014:
4011:
4008:
3985:
3982:
3979:
3976:
3973:
3970:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3912:
3909:
3906:
3886:
3883:
3880:
3857:
3837:
3817:
3814:
3811:
3808:
3805:
3802:
3780:
3776:
3772:
3769:
3749:
3746:
3742:
3738:
3735:
3732:
3728:
3705:
3701:
3697:
3694:
3674:
3671:
3668:
3665:
3662:
3659:
3654:
3650:
3640:. Notice that
3627:
3623:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3541:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3515:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3483:
3478:
3474:
3471:
3467:
3463:
3460:
3457:
3453:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3391:
3388:
3385:
3382:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3312:
3292:
3272:
3252:
3249:
3246:
3243:
3223:
3203:
3183:
3180:
3177:
3174:
3154:
3134:
3114:
3111:
3108:
3105:
3102:
3099:
3079:
3059:
3039:
3036:
3033:
3030:
3010:
3007:
3004:
2984:
2981:
2978:
2975:
2966:, we can keep
2955:
2935:
2912:
2892:
2872:
2869:
2866:
2863:
2860:
2857:
2835:
2831:
2827:
2824:
2804:
2801:
2797:
2793:
2790:
2787:
2783:
2760:
2756:
2752:
2749:
2729:
2726:
2723:
2720:
2717:
2714:
2709:
2705:
2687:
2683:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2601:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2575:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2543:
2538:
2534:
2531:
2527:
2523:
2520:
2517:
2513:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2451:
2448:
2445:
2442:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2372:
2352:
2332:
2312:
2309:
2306:
2303:
2294:can then make
2283:
2263:
2243:
2240:
2237:
2234:
2214:
2194:
2174:
2171:
2168:
2165:
2162:
2159:
2139:
2119:
2099:
2096:
2093:
2090:
2070:
2067:
2064:
2044:
2041:
2038:
2035:
2026:, we can keep
2015:
1995:
1972:
1952:
1932:
1929:
1926:
1923:
1900:
1880:
1860:
1840:
1820:
1800:
1797:
1794:
1791:
1788:
1785:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1727:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1655:
1652:
1650:
1647:
1632:
1610:
1607:
1604:
1601:
1598:
1595:
1575:
1555:
1552:
1549:
1546:
1543:
1540:
1520:
1517:
1514:
1511:
1508:
1505:
1484:
1480:
1477:
1455:
1451:
1447:
1444:
1441:
1438:
1435:
1408:
1405:
1384:
1381:
1378:
1375:
1372:
1369:
1364:
1359:
1356:
1353:
1348:
1324:
1319:
1316:
1313:
1308:
1286:
1283:
1280:
1277:
1257:
1254:
1251:
1248:
1228:
1225:
1222:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1160:
1159:
1145:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1063:
1041:
1038:
1035:
1032:
1015:
1003:
1000:
997:
994:
991:
988:
968:
965:
962:
959:
956:
953:
950:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
829:
826:
823:
820:
800:
797:
794:
791:
771:
743:
739:
736:
733:
730:
709:
688:
685:
682:
679:
676:
673:
670:
655:
652:
639:
619:
599:
579:
576:
573:
553:
550:
547:
544:
541:
538:
518:
515:
512:
509:
506:
503:
483:
480:
477:
474:
471:
468:
448:
445:
442:
439:
436:
427:continuous on
415:
390:
387:
386:
385:
378:
347:
344:
341:
338:
318:
315:
312:
309:
263:
231:
228:
225:
222:
219:
216:
196:
176:
155:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
84:
64:
61:
58:
55:
52:
32:
15:
9:
6:
4:
3:
2:
10303:
10292:
10289:
10287:
10284:
10282:
10279:
10278:
10276:
10267:
10263:
10260:
10256:
10255:
10250:
10245:
10240:
10239:
10234:
10231:
10226:
10223:
10219:
10216:
10214:
10210:
10207:
10204:
10201:
10200:
10192:
10189:
10188:
10177:
10173:
10168:
10160:
10156:
10151:
10146:
10142:
10138:
10131:
10123:
10117:
10113:
10112:
10104:
10096:
10092:
10087:
10082:
10078:
10074:
10070:
10063:
10055:
10051:
10047:
10043:
10039:
10035:
10031:
10024:
10016:
10014:9783319529561
10010:
10006:
10005:
9997:
9990:
9986:
9985:
9980:
9976:
9969:
9963:
9959:
9955:
9951:
9947:
9941:
9933:
9929:
9925:
9921:
9917:
9913:
9906:
9899:
9890:
9885:
9881:
9877:
9873:
9866:
9858:
9854:
9850:
9846:
9842:
9835:
9826:
9821:
9814:
9806:
9798:
9790:
9782:
9774:
9770:
9766:
9760:
9756:
9752:
9748:
9741:
9732:
9731:
9726:
9723:
9716:
9712:
9701:
9698:
9695:
9692:
9686:
9683:
9680:
9677:
9676:
9670:
9667:
9651:
9640:
9636:
9632:
9628:
9624:
9617:
9611:
9590:
9584:
9581:
9575:
9569:
9549:
9529:
9509:
9489:
9476:
9462:
9442:
9434:
9409:
9406:
9397:
9391:
9368:
9348:
9345:
9342:
9319:
9313:
9310:
9307:
9287:
9284:
9278:
9272:
9249:
9246:
9243:
9233:
9217:
9208:
9205:
9202:
9196:
9193:
9173:
9153:
9145:
9144:
9143:
9141:
9131:
9129:
9124:
9122:
9117:
9111:
9107:
9103:
9084:
9080:
9073:
9069:
9043:
9039:
9035:
9030:
9026:
9020:
9008:
9007:
8988:
8982:
8979:
8973:
8967:
8944:
8941:
8938:
8932:
8929:
8906:
8900:
8897:
8891:
8885:
8882:
8879:
8859:
8856:
8853:
8833:
8830:
8824:
8818:
8795:
8789:
8786:
8783:
8776:implies that
8760:
8754:
8751:
8748:
8745:
8739:
8733:
8725:
8724:
8704:
8698:
8695:
8689:
8683:
8660:
8654:
8646:
8645:
8625:
8622:
8619:
8613:
8610:
8602:
8601:
8591:
8587:
8573:
8570:
8564:
8558:
8535:
8532:
8529:
8523:
8520:
8494:
8488:
8485:
8479:
8473:
8464:
8461:
8458:
8449:
8443:
8440:
8434:
8428:
8402:
8374:
8371:
8368:
8323:
8320:
8317:
8311:
8308:
8295:
8284:
8281:
8278:
8277:is connected.
8261:
8255:
8247:
8231:
8211:
8205:
8202:
8199:
8179:
8159:
8149:
8145:
8144:
8139:
8115:
8112:
8109:
8101:
8098:
8095:
8092:
8089:
8083:
8080:
8077:
8074:
8071:
8068:
8043:
8040:
8032:
8010:
8004:
7997:subset, then
7996:
7980:
7977:
7974:
7954:
7948:
7945:
7942:
7934:
7933:metric spaces
7918:
7898:
7890:
7889:
7888:
7886:
7882:
7881:connectedness
7878:
7868:
7866:
7859:
7855:
7851:
7847:
7840:
7833:
7829:
7826:
7819:
7812:
7805:
7801:
7797:
7790:
7783:
7776:
7772:
7768:
7767:
7766:
7764:
7760:
7753:
7746:
7738:
7737:)=(0,...,0).
7736:
7732:
7728:
7724:
7720:
7712:
7705:
7695:
7688:
7684:
7674:
7670:
7667:
7660:
7656:
7652:
7645:
7638:
7631:
7627:
7623:
7622:
7621:
7619:
7615:
7611:
7607:
7603:
7593:
7589:
7585:
7575:
7571:
7567:
7563:
7559:
7556:-dimensional
7555:
7550:
7548:
7545:-dimensional
7544:
7540:
7525:
7522:
7520:
7516:
7512:
7507:
7505:
7495:
7491:
7486:
7480:
7473:
7466:
7459:
7455:
7451:
7444:
7438:
7435:
7431:
7427:
7408:
7404:
7397:
7393:
7368:
7358:
7356:
7352:
7347:
7343:
7339:
7335:
7331:
7315:
7306:
7289:
7283:
7280:
7274:
7268:
7248:
7228:
7208:
7185:
7179:
7176:
7170:
7164:
7141:
7135:
7132:
7126:
7120:
7100:
7080:
7060:
7057:
7054:
7045:
7043:
7039:
7035:
7025:
7023:
7019:
7015:
7001:
6998:
6992:
6986:
6966:
6963:
6957:
6951:
6931:
6928:
6922:
6916:
6896:
6889:
6885:
6869:
6866:
6863:
6843:
6823:
6803:
6800:
6797:
6794:
6791:
6771:
6768:
6762:
6756:
6734:
6731:
6726:
6723:
6720:
6717:
6697:
6694:
6688:
6682:
6659:
6653:
6650:
6644:
6638:
6635:
6629:
6623:
6620:
6614:
6608:
6605:
6582:
6576:
6573:
6562:
6556:
6553:
6547:
6541:
6513:
6510:
6507:
6504:
6501:
6498:
6495:
6489:
6486:
6466:
6463:
6460:
6440:
6437:
6431:
6425:
6422:
6419:
6399:
6396:
6390:
6384:
6364:
6361:
6355:
6349:
6329:
6322:
6318:
6302:
6299:
6296:
6276:
6256:
6236:
6233:
6230:
6210:
6207:
6204:
6201:
6198:
6178:
6175:
6169:
6163:
6140:
6137:
6134:
6130:
6125:
6122:
6100:
6097:
6092:
6089:
6086:
6083:
6063:
6060:
6054:
6048:
6025:
6019:
6016:
6013:
6002:
5996:
5993:
5987:
5981:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5929:
5926:
5906:
5903:
5900:
5877:
5871:
5868:
5865:
5862:
5859:
5839:
5836:
5830:
5824:
5804:
5801:
5795:
5789:
5769:
5766:
5760:
5754:
5751:
5748:
5745:
5739:
5733:
5710:
5704:
5695:
5678:
5669:
5666:
5659:
5643:
5620:
5617:
5614:
5608:
5605:
5585:
5565:
5562:
5559:
5539:
5536:
5530:
5524:
5501:
5498:
5492:
5486:
5483:
5477:
5474:
5471:
5465:
5462:
5456:
5453:
5430:
5427:
5424:
5418:
5415:
5395:
5392:
5386:
5380:
5357:
5351:
5348:
5345:
5342:
5336:
5330:
5307:
5301:
5298:
5295:
5292:
5286:
5280:
5260:
5257:
5251:
5245:
5242:
5236:
5230:
5210:
5207:
5201:
5195:
5192:
5186:
5180:
5171:
5154:
5148:
5145:
5142:
5139:
5133:
5127:
5104:
5098:
5095:
5092:
5089:
5083:
5077:
5063:
5049:
5046:
5040:
5034:
5014:
5011:
5008:
4988:
4985:
4982:
4979:
4973:
4967:
4964:
4961:
4958:
4955:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4909:
4906:
4902:
4895:
4892:
4886:
4880:
4860:
4840:
4820:
4817:
4812:
4809:
4805:
4781:
4778:
4775:
4772:
4769:
4763:
4758:
4755:
4751:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4704:
4700:
4693:
4690:
4684:
4678:
4658:
4635:
4632:
4629:
4626:
4623:
4617:
4612:
4608:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4560:
4557:
4537:
4534:
4528:
4522:
4519:
4513:
4507:
4504:
4501:
4498:
4492:
4486:
4461:
4457:
4453:
4448:
4444:
4434:
4431:
4408:
4405:
4402:
4396:
4388:
4384:
4380:
4377:
4374:
4369:
4365:
4361:
4358:
4335:
4332:
4327:
4323:
4296:
4293:
4290:
4264:
4261:
4258:
4252:
4249:
4240:
4226:
4223:
4212:
4206:
4203:
4197:
4191:
4176:
4172:
4168:
4160:
4157:
4154:
4126:
4123:
4120:
4114:
4111:
4088:
4085:
4080:
4076:
4052:
4032:
4012:
4009:
4006:
3997:
3983:
3980:
3974:
3968:
3945:
3942:
3939:
3933:
3930:
3910:
3907:
3904:
3884:
3881:
3878:
3869:
3855:
3835:
3815:
3812:
3806:
3800:
3778:
3774:
3770:
3767:
3747:
3744:
3736:
3733:
3730:
3703:
3699:
3695:
3692:
3669:
3666:
3663:
3657:
3652:
3648:
3625:
3621:
3617:
3611:
3608:
3602:
3599:
3596:
3593:
3590:
3561:
3558:
3555:
3549:
3543:
3535:
3532:
3526:
3520:
3517:
3506:
3500:
3497:
3491:
3485:
3472:
3469:
3461:
3458:
3455:
3427:
3424:
3421:
3415:
3412:
3389:
3386:
3383:
3360:
3357:
3354:
3351:
3345:
3339:
3336:
3333:
3324:
3310:
3290:
3270:
3263:greater than
3247:
3241:
3221:
3201:
3178:
3172:
3152:
3132:
3109:
3103:
3100:
3097:
3077:
3057:
3034:
3028:
3008:
3005:
3002:
2979:
2973:
2953:
2933:
2924:
2910:
2890:
2870:
2867:
2861:
2855:
2833:
2829:
2825:
2822:
2802:
2799:
2791:
2788:
2785:
2758:
2754:
2750:
2747:
2724:
2721:
2718:
2712:
2707:
2703:
2685:
2681:
2677:
2668:
2665:
2662:
2659:
2656:
2647:
2644:
2621:
2618:
2615:
2609:
2603:
2592:
2586:
2583:
2580:
2577:
2566:
2560:
2557:
2551:
2545:
2532:
2529:
2521:
2518:
2515:
2487:
2484:
2481:
2475:
2472:
2449:
2446:
2443:
2420:
2417:
2411:
2405:
2402:
2399:
2396:
2393:
2384:
2370:
2350:
2330:
2307:
2301:
2281:
2261:
2238:
2232:
2212:
2192:
2172:
2169:
2163:
2157:
2137:
2117:
2094:
2088:
2068:
2065:
2062:
2039:
2033:
2013:
1993:
1984:
1970:
1950:
1930:
1924:
1921:
1914:
1898:
1878:
1858:
1838:
1818:
1798:
1795:
1789:
1783:
1760:
1757:
1754:
1748:
1745:
1725:
1716:
1699:
1693:
1690:
1687:
1684:
1678:
1672:
1663:
1661:
1646:
1630:
1608:
1605:
1599:
1593:
1573:
1553:
1550:
1544:
1538:
1518:
1515:
1509:
1503:
1478:
1475:
1453:
1449:
1445:
1439:
1433:
1425:
1421:
1418:
1414:
1404:
1402:
1398:
1382:
1376:
1370:
1367:
1357:
1354:
1351:
1317:
1314:
1311:
1281:
1275:
1252:
1246:
1226:
1223:
1220:
1197:
1191:
1188:
1185:
1182:
1179:
1171:
1167:
1164:
1132:
1126:
1123:
1117:
1111:
1102:
1093:
1087:
1084:
1078:
1072:
1036:
1030:
1023:
1019:
1016:
1001:
998:
992:
986:
963:
960:
957:
951:
948:
928:
919:
913:
910:
904:
898:
889:
886:
883:
874:
868:
865:
859:
853:
824:
818:
795:
789:
769:
761:
758:
757:
756:
734:
731:
728:
683:
680:
677:
671:
668:
659:
651:
637:
617:
597:
577:
574:
571:
548:
542:
539:
536:
516:
513:
507:
501:
481:
478:
472:
466:
443:
440:
437:
426:
413:
404:
395:
383:
379:
376:
372:
368:
367:
366:
364:
359:
342:
336:
313:
307:
297:
293:
288:
285:contains the
284:
280:
277:
261:
253:
249:
229:
226:
220:
214:
194:
174:
166:
153:
129:
123:
120:
117:
114:
108:
102:
82:
59:
56:
53:
30:
21:
10262:Mizar system
10252:
10236:
10213:cut-the-knot
10205:at ProofWiki
10172:Keith Devlin
10167:
10140:
10136:
10130:
10110:
10103:
10076:
10072:
10062:
10037:
10033:
10023:
10007:. Springer.
10003:
9996:
9982:
9968:
9940:
9915:
9911:
9898:
9879:
9875:
9865:
9840:
9834:
9813:
9804:
9797:
9788:
9781:
9746:
9740:
9728:
9715:
9668:
9664:-dimensional
9642:
9634:
9630:
9626:
9622:
9615:
9609:
9481:
9430:
9137:
9125:
9115:
9109:
9105:
9101:
9082:
9078:
9071:
9067:
9041:
9037:
9033:
9018:
9010:
8721:
8642:
8598:
8596:
8589:
8293:
8286:
8282:
8151:
8141:
8135:
7884:
7874:
7862:
7857:
7853:
7849:
7842:
7835:
7831:
7821:
7814:
7807:
7803:
7799:
7792:
7791:)>0, and
7785:
7778:
7774:
7770:
7762:
7755:
7748:
7741:
7739:
7734:
7730:
7726:
7718:
7716:
7707:
7700:
7693:
7686:
7679:
7672:
7662:
7658:
7654:
7647:
7640:
7633:
7629:
7625:
7617:
7613:
7609:
7605:
7598:
7591:
7587:
7580:
7573:
7569:
7565:
7561:
7553:
7551:
7542:
7536:
7523:
7508:
7493:
7489:
7483:because the
7478:
7471:
7464:
7457:
7453:
7449:
7442:
7439:
7433:
7429:
7425:
7406:
7402:
7395:
7391:
7364:
7338:Simon Stevin
7307:
7046:
7031:
7017:
7016:
6887:
6883:
6320:
6316:
6289:. However,
5696:
5172:
5069:
4241:
3998:
3870:
3325:
2925:
2385:
1985:
1717:
1664:
1660:completeness
1657:
1419:
1410:
1400:
1396:
1165:
1162:
1161:
1017:
759:
660:
657:
406:
403:real numbers
400:
374:
360:
295:
291:
251:
245:
146:
8395:. Then, if
7877:topological
7447:defined by
7342:polynomials
6856:. However,
6317:upper bound
3050:by keeping
2995:within any
2110:by keeping
2055:within any
1018:Version II.
840:, that is,
610:moves from
363:corollaries
10275:Categories
10150:1701.02227
10079:: 107036.
9825:1704.00281
9707:References
9099:such that
9031:, and let
8551:such that
7879:notion of
7747:such that
7616:. Suppose
7456:) = sin(1/
7385:, and any
7261:such that
7113:such that
6479:such that
5919:such that
5517:. Because
4348:such that
4101:such that
3848:cannot be
3685:and every
3402:such that
2903:cannot be
2740:and every
2462:such that
1776:such that
1621:, because
1586:such that
1496:satisfies
1401:Version II
1166:Version II
979:such that
760:Version I.
389:Motivation
276:continuous
207:such that
10238:MathWorld
10095:0166-8641
10054:0166-8641
10040:: 40–46.
9773:132587955
9730:MathWorld
9582:−
9542:. Define
9410:ε
9343:ε
9215:→
8933:∈
8883:≠
8872:. Since
8857:∈
8846:for some
8787:∈
8524:∈
8378:→
8372::
8294:Version I
8209:→
8203::
8113:∈
8106:⟹
8078:∈
8044:⊂
8033:A subset
7995:connected
7978:⊂
7952:→
7946::
7856:in 1,...,
7773:in 1,...,
7729:on which
7628:in 1,...,
7558:simplices
7509:In fact,
7344:(using a
7316:φ
7284:φ
7249:β
7229:α
7186:β
7180:φ
7171:β
7142:α
7136:φ
7127:α
7101:β
7081:α
7061:φ
6732:δ
6727:−
6636:−
6606:−
6554:−
6514:δ
6502:δ
6499:−
6490:∈
6461:δ
6438:−
6420:ϵ
6362:≥
6234:∈
6138:−
6131:δ
6098:δ
6017:−
5994:−
5954:δ
5942:δ
5939:−
5930:∈
5901:δ
5869:−
5860:ϵ
5609:⊆
5578:so, that
5563:∈
5499:≤
5466:∈
5419:∈
5408:for some
5208:−
5120:, as the
5009:ε
4989:ε
4962:ε
4959:−
4933:ε
4930:−
4924:≥
4921:ε
4918:−
4910:∗
4907:∗
4813:∗
4810:∗
4782:δ
4764:∈
4759:∗
4756:∗
4728:ε
4716:ε
4705:∗
4671:, and so
4630:δ
4627:−
4618:∈
4613:∗
4585:δ
4573:δ
4570:−
4561:∈
4538:ε
4502:ε
4499:−
4458:δ
4445:δ
4432:δ
4397:⊆
4385:δ
4366:δ
4362:−
4324:δ
4320:∃
4312:is open,
4253:∈
4227:ε
4204:−
4183:⟹
4173:δ
4158:−
4115:∈
4109:∀
4077:δ
4073:∃
4007:ε
3999:Fix some
3934:∈
3908:≠
3882:≠
3771:∈
3748:δ
3734:−
3696:∈
3658:⊆
3603:δ
3600:−
3540:⟹
3533:−
3498:−
3477:⟹
3473:δ
3459:−
3416:∈
3410:∀
3384:δ
3381:∃
3352:−
3334:ε
3133:ε
3003:ε
2826:∈
2803:δ
2789:−
2751:∈
2713:⊆
2663:δ
2600:⟹
2584:−
2558:−
2537:⟹
2533:δ
2519:−
2476:∈
2470:∀
2444:δ
2441:∃
2403:−
2394:ε
2193:ε
2063:ε
1749:∈
1479:∈
1397:Version I
1368:⊆
1189:∈
1022:image set
952:∈
738:→
732::
9673:See also
9121:topology
9036: :
7769:For all
7723:interior
7624:For all
7445: :
7415:between
7389:between
7221:between
6529:implies
6115:, where
5694:exists.
5658:supremum
4833:because
4818:∉
4743:Picking
4550:for all
4025:. Since
3828:. Hence
3793:we have
3194:. Every
3090:. Since
2883:. Hence
2848:we have
2150:. Since
1913:supremum
1871:. Since
405:: given
287:interval
279:function
167:between
75:and let
10264:proof:
10174:(2007)
9932:2975545
9857:1800805
9021:, <)
8248:, then
7827:)<0.
7721:in the
7474:(0) = 0
7028:History
7018:Remark:
6749:, then
6156:, then
5173:Define
3373:. Then
3326:Choose
2433:. Then
2386:Choose
1811:. Then
1163:Remark:
755:. Then
654:Theorem
10118:
10093:
10052:
10011:
9930:
9855:
9771:
9761:
9562:to be
9006:Q.E.D.
8290:
8087:
7586:. Let
7564:be an
7560:. Let
7467:> 0
4424:. Set
4242:Since
590:while
283:domain
281:whose
250:, the
10145:arXiv
10143:(3).
9928:JSTOR
9908:(PDF)
9820:arXiv
9769:S2CID
9482:Take
9023:is a
8811:, or
8594:Proof
8244:is a
7993:is a
7699:,...,
7678:,...,
7597:,...,
7579:,...,
7485:limit
7423:with
7346:cubic
6884:least
6342:, so
6223:, so
3871:With
2254:. No
1649:Proof
1213:with
382:image
298:]
290:[
274:is a
10116:ISBN
10091:ISSN
10050:ISSN
10009:ISBN
9962:link
9960:See
9759:ISBN
9629:) =
9522:and
9407:<
9346:>
9311:<
9300:and
9285:<
9166:and
9146:Let
9126:The
9108:) =
9089:<
9076:and
9057:and
9049:and
8980:<
8752:<
8746:<
8723:(**)
8696:<
8600:(**)
8465:<
8459:<
8172:and
8140:and
8099:<
8093:<
7931:are
7911:and
7547:cube
7537:The
7469:and
7462:for
7432:) =
7419:and
7400:and
7377:and
7241:and
7177:>
7157:and
7133:<
7093:and
7047:Let
6929:>
6867:<
6801:<
6795:<
6784:and
6769:>
6695:>
6651:<
6621:<
6574:<
6464:>
6397:>
6300:>
6208:<
6202:<
6191:and
6176:<
6126:>
6061:<
6014:<
5904:>
5837:<
5782:and
5767:>
5746:<
5537:<
5349:<
5343:<
5299:<
5293:<
5146:>
5140:>
5096:<
5090:<
5012:>
4980:<
4965:<
4893:>
4719:<
4691:<
4520:<
4505:<
4333:>
4280:and
4224:<
4169:<
4086:>
4010:>
3897:and
3813:>
3745:<
3556:>
3518:<
3470:<
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3358:>
3165:and
3101:<
3006:>
2868:<
2800:<
2616:<
2578:<
2530:<
2447:>
2418:>
2225:and
2170:<
2066:>
1796:<
1718:Let
1691:<
1685:<
1531:and
1468:for
1268:and
1224:<
1020:the
890:<
884:<
811:and
494:and
380:The
371:root
329:and
187:and
121:<
115:<
10211:at
10155:doi
10081:doi
10077:275
10042:doi
10038:202
9954:doi
9920:doi
9884:doi
9845:doi
9751:doi
9618:= 0
9138:In
9095:in
8644:(*)
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8423:min
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8143:(*)
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7685:on
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7481:= 0
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4438:min
3585:max
3021:of
2946:at
2651:min
2363:in
2081:of
2006:at
1928:sup
1170:set
1106:max
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893:max
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762:if
630:to
377:).
246:In
10277::
10251:.
10235:.
10153:.
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10139:.
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9926:.
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9914:.
9910:.
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9727:.
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8130:**
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7506:.
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9635:B
9633:(
9631:f
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9623:f
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9610:d
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9594:)
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9585:f
9579:)
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9422:.
9404:|
9401:)
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9395:(
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9320:b
9317:(
9314:f
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9288:0
9282:)
9279:a
9276:(
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9253:]
9250:b
9247:,
9244:a
9241:[
9218:R
9212:]
9209:b
9206:,
9203:a
9200:[
9197::
9194:f
9174:b
9154:a
9116:R
9110:u
9106:c
9104:(
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9097:X
9093:c
9085:)
9083:b
9081:(
9079:f
9074:)
9072:a
9070:(
9068:f
9063:Y
9059:u
9055:X
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9034:f
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9013:X
8992:)
8989:a
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8901:f
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8895:)
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8886:f
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8860:I
8854:c
8834:u
8831:=
8828:)
8825:c
8822:(
8819:f
8799:)
8796:I
8793:(
8790:f
8784:u
8764:)
8761:b
8758:(
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8743:)
8740:a
8737:(
8734:f
8708:)
8705:b
8702:(
8699:f
8693:)
8690:a
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8664:)
8661:I
8658:(
8655:f
8629:]
8626:b
8623:,
8620:a
8617:[
8614:=
8611:I
8574:u
8571:=
8568:)
8565:c
8562:(
8559:f
8539:)
8536:b
8533:,
8530:a
8527:(
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8501:)
8498:)
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8480:a
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8471:(
8462:u
8456:)
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8450:b
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8441:,
8438:)
8435:a
8432:(
8429:f
8426:(
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8348:R
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8324:b
8321:,
8318:a
8315:[
8312:=
8309:I
8296:)
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8265:)
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8259:(
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8232:X
8212:Y
8206:X
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8180:Y
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8132:)
8116:E
8110:r
8102:y
8096:r
8090:x
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8075:y
8072:,
8069:x
8048:R
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8028:*
8014:)
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7899:X
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7850:x
7848:(
7845:i
7843:f
7838:0
7836:v
7832:x
7824:0
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7820:(
7817:i
7815:f
7810:i
7808:v
7804:x
7800:x
7798:(
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7793:f
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7779:f
7775:n
7771:i
7763:i
7758:i
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7754:(
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7749:f
7744:i
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7735:z
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7713:.
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7682:n
7680:f
7676:1
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7668:;
7665:i
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7650:i
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7639:(
7636:i
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7592:f
7588:F
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7554:n
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7492:(
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7479:x
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7465:x
7460:)
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7450:f
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7407:b
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7398:)
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7383:f
7379:b
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7293:)
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7281:=
7278:)
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7174:)
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7139:(
7130:)
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7058:,
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7002:u
6999:=
6996:)
6993:c
6990:(
6987:f
6967:0
6964:=
6961:)
6958:c
6955:(
6952:g
6932:0
6926:)
6923:c
6920:(
6917:g
6897:c
6870:c
6864:x
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6804:c
6798:x
6792:a
6772:0
6766:)
6763:x
6760:(
6757:g
6735:2
6724:c
6721:=
6718:x
6698:0
6692:)
6689:x
6686:(
6683:g
6663:)
6660:c
6657:(
6654:g
6648:)
6645:c
6642:(
6639:g
6633:)
6630:x
6627:(
6624:g
6618:)
6615:c
6612:(
6609:g
6586:)
6583:c
6580:(
6577:g
6570:|
6566:)
6563:c
6560:(
6557:g
6551:)
6548:x
6545:(
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6517:)
6511:+
6508:c
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6496:c
6493:(
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6467:0
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6435:)
6432:c
6429:(
6426:g
6423:=
6400:0
6394:)
6391:c
6388:(
6385:g
6365:0
6359:)
6356:c
6353:(
6350:g
6330:c
6303:c
6297:x
6277:S
6257:x
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6231:x
6211:b
6205:x
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6173:)
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6167:(
6164:g
6141:c
6135:b
6123:N
6101:N
6093:+
6090:c
6087:=
6084:x
6064:0
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6055:x
6052:(
6049:g
6029:)
6026:c
6023:(
6020:g
6010:|
6006:)
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6000:(
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5991:)
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5985:(
5982:g
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5951:+
5948:c
5945:,
5936:c
5933:(
5927:x
5907:0
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5878:c
5875:(
5872:g
5866:0
5863:=
5840:0
5834:)
5831:c
5828:(
5825:g
5805:0
5802:=
5799:)
5796:c
5793:(
5790:g
5770:0
5764:)
5761:c
5758:(
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5752:,
5749:0
5743:)
5740:c
5737:(
5734:g
5714:)
5711:c
5708:(
5705:g
5682:)
5679:S
5676:(
5670:=
5667:c
5644:S
5624:]
5621:b
5618:,
5615:a
5612:[
5606:S
5586:S
5566:S
5560:a
5540:0
5534:)
5531:a
5528:(
5525:g
5505:}
5502:0
5496:)
5493:x
5490:(
5487:g
5484::
5481:]
5478:b
5475:,
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5469:[
5463:x
5460:{
5457:=
5454:S
5434:]
5431:b
5428:,
5425:a
5422:[
5416:c
5396:0
5393:=
5390:)
5387:c
5384:(
5381:g
5361:)
5358:b
5355:(
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5346:0
5340:)
5337:a
5334:(
5331:g
5311:)
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5305:(
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5296:u
5290:)
5287:a
5284:(
5281:f
5261:u
5258:+
5255:)
5252:x
5249:(
5246:g
5243:=
5240:)
5237:x
5234:(
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5211:u
5205:)
5202:x
5199:(
5196:f
5193:=
5190:)
5187:x
5184:(
5181:g
5158:)
5155:b
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5143:u
5137:)
5134:a
5131:(
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5108:)
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5102:(
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5087:)
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5081:(
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5047:=
5044:)
5041:c
5038:(
5035:f
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4977:)
4974:c
4971:(
4968:f
4956:u
4936:.
4927:u
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4887:c
4884:(
4881:f
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4821:S
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4773:,
4770:c
4767:(
4752:a
4731:.
4725:+
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4682:(
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4639:]
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3653:2
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3615:]
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