6222:
6112:
6221:
6111:
4307:
3796:
2672:
1978:
733:. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between
6104:
around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of
6344:
in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of
Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed
5116:
subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the
6715:. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form
4155:
3644:
2520:
1826:
8967:
possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact
Hausdorff space to a uniform space is uniformly continuous.
8338:
formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous.
6325:
around that point, there is a function value directly above or below the rectangle. There might be a graph point where the graph is completely inside the height of the rectangle but this is not true for every point of the
4549:
4881:
1724:
7999:
5392:
7095:
20:
4482:
4397:
2885:
62:
6999:
4793:
4990:
446:
are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number
2951:
2452:
2386:
1401:
1295:
1229:
103:
7751:
6105:
the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)
4028:
2191:
4663:
6472:
4118:
2011:
5250:
4719:
869:
Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of
544:
8613:
6870:
3006:
2776:
2320:
1163:
4147:
3245:, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an
2040:
782:
147:
8935:
5308:
6251:
6141:
5851:
5016:
2220:
1576:
1057:
8546:
8259:
7841:
7653:
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7217:
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6713:
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5636:
3902:
365:
5730:
4089:
2098:
2069:
8851:
6300:
6213:
6079:
5443:
6560:
6277:
6167:
5955:
5877:
5816:
5330:
5036:
4595:
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3159:
3119:
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2715:
2246:
1602:
1083:
711:
671:
233:
5674:
5205:
3992:
1752:
3842:
856:
3545:
3506:
996:
957:
651:
464:
8710:
8650:
8420:
6323:
6190:
6102:
6056:
4302:{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;\forall y\in I:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon }
3791:{\displaystyle \forall x\in I\;\forall \varepsilon >0\;\exists \delta >0\;\forall y\in I:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon }
3464:
2667:{\displaystyle \forall x\in X\;\forall \varepsilon >0\;\exists \delta >0\;\forall y\in X:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon }
1973:{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in X\;\forall y\in X:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon }
1634:
1115:
915:
8116:
6493:
6015:
5056:
4924:
4615:
4420:
4355:
3139:
3039:
691:
631:
604:
484:
444:
424:
273:
7802:
6613:
2822:
2741:
2272:
6924:
6897:
6740:
6682:
6587:
5757:
3938:
3322:
1546:
1519:
814:
6340:
in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by
5935:
5906:
322:
5557:
173:
8955:
8871:
8804:
8782:
8760:
8738:
8500:
8480:
8460:
8440:
8388:
8368:
8324:
8302:
8280:
8223:
8201:
8179:
8157:
8136:
8089:
8067:
8045:
8023:
7907:
7887:
7862:
7772:
7699:
7674:
7620:
7598:
7576:
7554:
7532:
7511:
7490:
7433:
7413:
7392:
7371:
7351:
7330:
7305:
7284:
7259:
7239:
7185:
7163:
7143:
7123:
6800:
6760:
6653:
6633:
6537:
6515:
6407:
6381:
5995:
5975:
5697:
5270:
5173:
5096:
5076:
4904:
4440:
4331:
3633:
3613:
3566:
3429:
3409:
3389:
3369:
3342:
3291:
3271:
3243:
3223:
3203:
3179:
3079:
3059:
2971:
2796:
2695:
2515:
2492:
2472:
2154:
2121:
1821:
1796:
1776:
1488:
1468:
1441:
1421:
1335:
1315:
1024:
731:
584:
564:
400:
293:
253:
213:
193:
5510:
5478:
7753:
is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like
4499:
4801:
5515:
If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval.
1639:
7915:
295:
as the window moves along it over its domain, then that window's width would need to be infinitesimally small (nonreal), meaning that
5335:
7775:, uniform continuity is a rather strong condition. It is desirable to have a weaker condition from which to deduce extendability.
9174:
7010:
3344:), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see
4448:
4363:
5332:
that vanish at infinity, is uniformly continuous. This is a generalization of the Heine-Cantor theorem mentioned above, since
9135:
2827:
26:
6932:
4724:
4929:
7415:, and this has the further pleasant consequence that if the extension exists, it is unique. A sufficient condition for
2890:
2391:
2325:
1340:
1234:
1168:
67:
9087:
9068:
9036:
7705:
3997:
8960:
In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences.
2162:
4620:
9184:
6413:
6341:
5732:
is continuous everywhere on the real line but is not uniformly continuous on the line, since its derivative is
4094:
3249:, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a
1987:
5210:
4668:
489:
9108:
8551:
6834:
2976:
2746:
740:
Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as
2277:
1120:
8334:
A typical application of the extendability of a uniformly continuous function is the proof of the inverse
4123:
2016:
743:
108:
9103:
8876:
5275:
1444:
870:
9098:
7866:
7468:
6230:
6120:
5830:
4995:
2199:
1555:
1036:
8505:
8232:
7807:
7626:
7438:
7190:
6805:
6687:
5762:
5590:
3847:
331:
9189:
8348:
7680:
It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to
5702:
4065:
2074:
2045:
8810:
7308:
6282:
6195:
6061:
5413:
5129:
6543:
6256:
6169:
such that when we draw a rectangle around each point of the graph with a width slightly less than
6146:
5940:
5856:
5795:
5313:
5021:
4580:
4554:
4036:
3572:
3144:
3104:
3084:
2700:
2225:
1581:
1062:
696:
656:
218:
8964:
5645:
5178:
3943:
1729:
6279:
there is a point on the graph so that when we draw a rectangle with a height slightly less than
3805:
819:
8335:
5570:
5143:
3636:
3511:
3472:
3324:(the characteristics of which at nonstandard points are determined by the global properties of
3246:
1981:
962:
923:
636:
449:
380:
8683:
8623:
8393:
6305:
6172:
6084:
6020:
3437:
1607:
1088:
888:
8652:, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in
8618:
8095:
6763:
6478:
6359:
6000:
5113:
5102:
5041:
4909:
4600:
4405:
4340:
3124:
3024:
2124:
676:
616:
589:
469:
429:
409:
258:
7781:
6592:
2801:
2720:
2251:
633:(the size of a function domain interval over which function value differences are less than
8717:
6902:
6875:
6718:
6658:
6565:
5735:
5563:
5519:
5038:. This means that there is no specifiable (no matter how small it is) positive real number
3907:
3300:
3273:, in the sense that the standard definition of uniform continuity refers to every point of
1755:
1524:
1497:
859:
787:
5911:
5882:
5523:
298:
275:. If there existed a window whereof top and/or bottom is never penetrated by the graph of
8:
8983:
8977:
8653:
5536:
4493:
3018:
610:
152:
8940:
8856:
8789:
8767:
8745:
8723:
8485:
8465:
8445:
8425:
8373:
8353:
8309:
8287:
8265:
8208:
8186:
8164:
8142:
8121:
8074:
8052:
8030:
8008:
7892:
7872:
7847:
7757:
7684:
7659:
7605:
7583:
7561:
7539:
7517:
7496:
7475:
7418:
7398:
7377:
7356:
7336:
7315:
7290:
7269:
7244:
7224:
7170:
7148:
7128:
7108:
6785:
6745:
6638:
6618:
6615:. Uniform continuity can be expressed as the condition that (the natural extension of)
6522:
6500:
6392:
6366:
5980:
5960:
5682:
5255:
5158:
5081:
5061:
4889:
4425:
4316:
4059:
3618:
3598:
3551:
3414:
3394:
3374:
3354:
3327:
3276:
3256:
3228:
3208:
3188:
3164:
3064:
3044:
2956:
2781:
2680:
2500:
2477:
2457:
2139:
2106:
1806:
1781:
1761:
1549:
1473:
1453:
1426:
1406:
1320:
1300:
1009:
716:
569:
549:
385:
278:
238:
198:
178:
5483:
5451:
19:
9179:
9131:
9120:
9083:
9064:
9032:
8669:
7865:(assuming the existence of qth roots of positive real numbers, an application of the
6775:
9047:
9154:
9020:
5584:
5583:
Functions that are unbounded on a bounded domain are not uniformly continuous. The
5149:
5122:
5105:
function (over a compact interval) is uniformly continuous. On the other hand, the
195:
penetrates the (interior of the) top and/or bottom of that window. This means that
998:, the following definitions of uniform continuity and (ordinary) continuity hold.
8328:
6385:
6227:
For functions that are not uniformly continuous, there is a positive real number
5408:
5118:
5106:
4496:, but the converse does not hold. Consider for instance the continuous function
5530:
2474:. Thus, (ordinary) continuity is a local property of the function at the point
9168:
9158:
8677:
5522:
map between two metric spaces is uniformly continuous. More generally, every
3467:
2697:
is said to be continuous if there is a function of all positive real numbers
918:
874:
734:
9115:
8657:
6337:
4544:{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto x^{2}}
5559:. This shows uniformly continuous functions are not always differentiable.
9127:
6345:
interval is uniformly continuous, but he does not give a complete proof.
5135:
4876:{\displaystyle f\left(x+\delta \right)-f(x)=2x\cdot \delta +\delta ^{2},}
4574:
4031:
2194:
1031:
376:
6655:, but at all points in its non-standard counterpart (natural extension)
426:
such that function values over any function domain interval of the size
9145:
Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity",
873:
of distinct points, so it requires a metric space, or more generally a
6058:
of the graph, if we draw a rectangle with a height slightly less than
5152:
of continuous functions follows almost immediately from this theorem.
4597:, uniform continuity requires the existence of a positive real number
5699:
grows large cannot be uniformly continuous. The exponential function
1491:
816:, or if their slopes become unbounded on an infinite domain, such as
5827:
For a uniformly continuous function, for every positive real number
3012:
1719:{\displaystyle |x-y|<\delta \implies |f(x)-f(y)|<\varepsilon }
863:
6117:
For uniformly continuous functions, for each positive real number
4313:
Thus for continuity on the interval, one takes an arbitrary point
4058:
For uniform continuity, the order of the first, second, and third
3017:
In the definitions, the difference between uniform continuity and
7994:{\displaystyle f(x+\delta )-f(x)=a^{x}\left(a^{\delta }-1\right)}
6774:
For a function between metric spaces, uniform continuity implies
5642:
uniformly continuous on that interval, as it goes to infinity as
3293:. On the other hand, it is possible to give a definition that is
2778:
representing the maximum positive real number, such that at each
862:
between metric spaces is uniformly continuous, in particular any
8668:
Just as the most natural and general setting for continuity is
6215:, the graph lies completely inside the height of the rectangle.
673:, while in (ordinary) continuity there is a locally applicable
8342:
7843:
can be given a precise definition only for rational values of
5387:{\displaystyle C_{c}(\mathbb {R} )\subset C_{0}(\mathbb {R} )}
3021:
is that, in uniform continuity there is a globally applicable
613:
is that, in uniform continuity there is a globally applicable
8997:
5533:
is uniformly continuous, despite not being differentiable at
5146:
of the real line, it is uniformly continuous on that interval
6336:
The first published definition of uniform continuity was by
5445:
are the simplest examples of uniformly continuous functions.
880:
1552:, yielding the following definition: for every real number
9043:
Chapter II is a comprehensive reference of uniform spaces.
8048:
of all rational numbers; however for any bounded interval
7090:{\displaystyle \lim _{n\to \infty }|f(x_{n})-f(y_{n})|=0.}
7804:
is a real number. At the precalculus level, the function
7373:: that is, we may assume without loss of generality that
6872:
is uniformly continuous then for every pair of sequences
5272:
is uniformly continuous. In particular, every element of
4477:{\displaystyle \cdots \exists \delta \,\forall x\cdots .}
4392:{\displaystyle \cdots \forall x\,\exists \delta \cdots ,}
609:
The difference between uniform continuity and (ordinary)
8672:, the most natural and general setting for the study of
8118:
is uniformly continuous, hence Cauchy-continuous, hence
7100:
5109:
is uniformly continuous but not absolutely continuous.
3061:
over which values of the metric for function values in
2131:
763:
128:
8980: – Function reducing distance between all points
8943:
8879:
8859:
8813:
8792:
8770:
8748:
8726:
8686:
8626:
8554:
8508:
8488:
8468:
8448:
8428:
8396:
8376:
8356:
8312:
8290:
8268:
8235:
8211:
8189:
8167:
8145:
8124:
8098:
8077:
8055:
8033:
8011:
7918:
7895:
7875:
7850:
7810:
7784:
7760:
7708:
7687:
7662:
7629:
7608:
7586:
7564:
7542:
7520:
7499:
7478:
7441:
7421:
7401:
7380:
7359:
7339:
7318:
7293:
7272:
7247:
7227:
7193:
7173:
7151:
7131:
7111:
7013:
6935:
6905:
6878:
6837:
6808:
6788:
6748:
6721:
6690:
6661:
6641:
6621:
6595:
6568:
6546:
6525:
6503:
6481:
6416:
6395:
6369:
6308:
6285:
6259:
6233:
6198:
6175:
6149:
6123:
6087:
6064:
6023:
6003:
5983:
5963:
5943:
5914:
5885:
5859:
5833:
5798:
5765:
5738:
5705:
5685:
5648:
5593:
5539:
5486:
5454:
5416:
5338:
5316:
5278:
5258:
5213:
5181:
5161:
5084:
5064:
5044:
5024:
4998:
4932:
4912:
4892:
4804:
4727:
4671:
4623:
4603:
4583:
4557:
4502:
4451:
4428:
4408:
4366:
4343:
4319:
4158:
4126:
4097:
4068:
4039:
4000:
3946:
3910:
3850:
3808:
3647:
3621:
3601:
3575:
3554:
3514:
3475:
3440:
3417:
3397:
3377:
3357:
3330:
3303:
3279:
3259:
3231:
3211:
3191:
3167:
3147:
3127:
3107:
3087:
3067:
3047:
3027:
2979:
2959:
2893:
2880:{\displaystyle d_{1}(x,y)<\delta (\varepsilon ,x)}
2830:
2804:
2784:
2749:
2723:
2703:
2683:
2523:
2503:
2480:
2460:
2394:
2328:
2280:
2254:
2228:
2202:
2165:
2142:
2109:
2077:
2048:
2019:
1990:
1829:
1809:
1784:
1764:
1732:
1642:
1610:
1584:
1558:
1527:
1500:
1476:
1456:
1429:
1409:
1343:
1323:
1303:
1237:
1171:
1123:
1091:
1065:
1039:
1012:
965:
926:
891:
822:
790:
746:
719:
699:
679:
659:
639:
619:
592:
572:
552:
492:
472:
452:
432:
412:
388:
334:
301:
281:
261:
241:
221:
201:
181:
155:
111:
70:
57:{\displaystyle 2\varepsilon \in \mathbb {R} _{>0}}
29:
3431:
are structurally similar as shown in the following.
8663:
6994:{\displaystyle \lim _{n\to \infty }|x_{n}-y_{n}|=0}
4788:{\displaystyle |f(x_{1})-f(x_{2})|<\varepsilon }
1445:
the definition of a neighbourhood in a metric space
1001:
23:As the center of the blue window, with real height
9144:
9119:
9082:. Graduate Texts in Mathematics. Springer-Verlag.
9003:
8949:
8929:
8865:
8845:
8798:
8776:
8754:
8732:
8704:
8644:
8607:
8540:
8494:
8474:
8454:
8434:
8414:
8382:
8362:
8318:
8296:
8274:
8253:
8217:
8195:
8173:
8151:
8130:
8110:
8083:
8061:
8039:
8017:
7993:
7901:
7881:
7856:
7835:
7796:
7766:
7745:
7693:
7668:
7647:
7614:
7592:
7570:
7548:
7526:
7505:
7484:
7459:
7427:
7407:
7386:
7365:
7345:
7324:
7299:
7278:
7253:
7233:
7211:
7179:
7157:
7137:
7117:
7089:
6993:
6918:
6891:
6864:
6823:
6794:
6754:
6734:
6707:
6676:
6647:
6627:
6607:
6581:
6554:
6531:
6509:
6487:
6466:
6401:
6375:
6317:
6294:
6271:
6245:
6207:
6184:
6161:
6135:
6096:
6073:
6050:
6009:
5989:
5969:
5949:
5929:
5900:
5871:
5845:
5810:
5784:
5751:
5724:
5691:
5668:
5630:
5551:
5504:
5472:
5437:
5386:
5324:
5302:
5264:
5244:
5199:
5167:
5090:
5070:
5050:
5030:
5010:
4985:{\displaystyle |f(x+\beta )-f(x)|<\varepsilon }
4984:
4918:
4898:
4875:
4787:
4713:
4657:
4609:
4589:
4565:
4543:
4476:
4434:
4414:
4391:
4349:
4325:
4301:
4141:
4112:
4083:
4047:
4022:
3986:
3932:
3896:
3836:
3790:
3627:
3607:
3587:
3560:
3539:
3500:
3458:
3423:
3403:
3383:
3363:
3336:
3316:
3285:
3265:
3237:
3217:
3197:
3173:
3153:
3133:
3121:while in continuity there is a locally applicable
3113:
3093:
3073:
3053:
3033:
3000:
2965:
2945:
2879:
2816:
2790:
2770:
2735:
2709:
2689:
2666:
2509:
2486:
2466:
2446:
2380:
2314:
2266:
2240:
2214:
2185:
2148:
2115:
2092:
2063:
2034:
2005:
1972:
1815:
1790:
1770:
1746:
1718:
1628:
1596:
1570:
1540:
1513:
1482:
1462:
1435:
1415:
1395:
1329:
1309:
1289:
1223:
1157:
1109:
1077:
1051:
1018:
990:
951:
909:
850:
808:
776:
725:
705:
685:
665:
645:
625:
598:
578:
558:
538:
478:
458:
438:
418:
394:
359:
316:
287:
267:
247:
227:
207:
187:
167:
141:
97:
56:
8656:to extend a linear map off a dense subspace of a
7702:. The converse does not hold, since the function
3013:Local continuity versus global uniform continuity
9166:
7015:
6937:
5679:Functions whose derivative tends to infinity as
5215:
2946:{\displaystyle d_{2}(f(x),f(y))<\varepsilon }
2447:{\displaystyle \{y\in X:d_{1}(x,y)<\delta \}}
2381:{\displaystyle d_{2}(f(x),f(y))<\varepsilon }
1396:{\displaystyle \{x\in X:d_{1}(x,y)<\delta \}}
1290:{\displaystyle \{y\in X:d_{1}(x,y)<\delta \}}
1224:{\displaystyle d_{2}(f(x),f(y))<\varepsilon }
215:ranges over an interval larger than or equal to
98:{\displaystyle 2\delta \in \mathbb {R} _{>0}}
7746:{\displaystyle f:R\rightarrow R,x\mapsto x^{2}}
7241:be extended to a continuous function on all of
7311:. So it is necessary and sufficient to extend
6635:is microcontinuous not only at real points in
4023:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
8261:whose restriction to every bounded subset of
7219:a continuous function. A question to answer:
3205:is continuous, or not, at a particular point
3185:property of a function — that is, a function
8390:, the notion of uniform continuity of a map
2441:
2395:
2186:{\displaystyle {\underline {{\text{at }}x}}}
1390:
1344:
1284:
1238:
175:, there comes a point at which the graph of
16:Uniform restraint of the change in functions
9096:
9058:
8343:Generalization to topological vector spaces
6779:
5397:
4658:{\displaystyle x_{1},x_{2}\in \mathbb {R} }
3008:is a monotonically non-decreasing function.
1550:standard one-dimensional Euclidean distance
8986: – Uniformly continuous homeomorphism
6348:
5562:Despite being nowhere differentiable, the
4197:
4184:
4171:
3686:
3673:
3660:
3635:) is expressed by a formula starting with
3351:A mathematical definition that a function
3345:
2562:
2549:
2536:
1868:
1855:
1842:
1740:
1736:
1672:
1668:
9046:
8283:is uniformly continuous is extendable to
6852:
6811:
6701:
6548:
6467:{\displaystyle f^{*}(a+\delta )-f^{*}(a)}
6253:such that for every positive real number
5573:set of functions is uniformly continuous.
5377:
5353:
5318:
5293:
4651:
4559:
4518:
4510:
4461:
4376:
4249:
4245:
4213:
4113:{\displaystyle \forall \varepsilon >0}
4041:
4016:
4008:
3738:
3734:
3702:
2614:
2610:
2578:
2006:{\displaystyle \forall \varepsilon >0}
1920:
1916:
1884:
881:Definition for functions on metric spaces
82:
41:
9019:
7513:is complete (and thus the completion of
7493:of a Cauchy sequence remains Cauchy. If
6353:
5448:Any continuous function on the interval
5245:{\displaystyle \lim _{x\to \infty }f(x)}
4714:{\displaystyle |x_{1}-x_{2}|<\delta }
858:on the real (number) line. However, any
539:{\displaystyle |f(x)-f(y)|<\epsilon }
18:
8608:{\displaystyle f(v_{1})-f(v_{2})\in B.}
8026:is not uniformly continuous on the set
7535:), then every continuous function from
6865:{\displaystyle f:A\to \mathbb {R} ^{n}}
6589:is microcontinuous at every real point
5310:, the space of continuous functions on
5121:to the integers endowed with the usual
4926:needs to be lower and lower to satisfy
4492:Every uniformly continuous function is
4402:while for uniform continuity, a single
3001:{\displaystyle \delta (\varepsilon ,x)}
2771:{\displaystyle \delta (\varepsilon ,x)}
2123:is uniformly continuous if it admits a
466:, then there is a positive real number
9167:
9077:
8229:More generally, a continuous function
8182:, there is then a unique extension of
4906:goes to be a higher and higher value,
1823:is said to be uniformly continuous if
9114:
7579:is Cauchy-continuous. Therefore when
4337:and then there must exist a distance
2315:{\displaystyle d_{1}(x,y)<\delta }
1158:{\displaystyle d_{1}(x,y)<\delta }
586:in any function interval of the size
8138:extends to a continuous function on
7101:Relations with the extension problem
6769:
5480:is also uniformly continuous, since
4142:{\displaystyle \exists \delta >0}
2035:{\displaystyle \exists \delta >0}
777:{\displaystyle f(x)={\tfrac {1}{x}}}
142:{\displaystyle f(x)={\tfrac {1}{x}}}
9122:Principles of Mathematical Analysis
8930:{\displaystyle (f(x_{1}),f(x_{2}))}
8204:to a continuous function on all of
7435:to extend to a continuous function
5303:{\displaystyle C_{0}(\mathbb {R} )}
4422:must work uniformly for all points
2132:Definition of (ordinary) continuity
406:if there is a positive real number
328:uniformly continuous. The function
13:
9013:
7025:
6947:
5805:
5779:
5225:
5191:
4462:
4455:
4377:
4370:
4198:
4185:
4172:
4159:
4127:
4098:
4069:
3687:
3674:
3661:
3648:
3297:in terms of the natural extension
2563:
2550:
2537:
2524:
2078:
2049:
2020:
1991:
1869:
1856:
1843:
1830:
14:
9201:
8712:between uniform spaces is called
8160:. But since this holds for every
7623:extends to a continuous function
6246:{\displaystyle \varepsilon >0}
6136:{\displaystyle \varepsilon >0}
5846:{\displaystyle \varepsilon >0}
5526:function is uniformly continuous.
5142:if a function is continuous on a
5011:{\displaystyle \beta <\delta }
2215:{\displaystyle \varepsilon >0}
1571:{\displaystyle \varepsilon >0}
1052:{\displaystyle \varepsilon >0}
8664:Generalization to uniform spaces
8541:{\displaystyle v_{1}-v_{2}\in A}
8254:{\displaystyle f:S\rightarrow R}
7889:to a function defined on all of
7836:{\displaystyle f:x\mapsto a^{x}}
7648:{\displaystyle f:X\rightarrow R}
7460:{\displaystyle f:X\rightarrow R}
7212:{\displaystyle f:S\rightarrow R}
6824:{\displaystyle \mathbb {R} ^{n}}
6766:for more details and examples).
6708:{\displaystyle ^{*}\mathbb {R} }
6220:
6192:and a height slightly less than
6143:there is a positive real number
6110:
5997:are within the maximum distance
5853:there is a positive real number
5822:
5785:{\displaystyle e^{x}\to \infty }
5631:{\displaystyle (-\pi /2,\pi /2)}
3897:{\displaystyle d_{2}(f(x),f(y))}
3041:(the size of a neighbourhood in
1002:Definition of uniform continuity
360:{\displaystyle g(x)={\sqrt {x}}}
6302:and a width slightly less than
6081:and width a slightly less than
5134:every continuous function on a
4577:. Given a positive real number
9175:Theory of continuous functions
9052:Foundations of Modern Analysis
9025:General Topology: Chapters 1–4
9004:Rusnock & Kerr-Lawson 2005
8924:
8921:
8908:
8899:
8886:
8880:
8840:
8814:
8696:
8636:
8593:
8580:
8571:
8558:
8462:, there exists a neighborhood
8422:becomes: for any neighborhood
8406:
8245:
7949:
7943:
7934:
7922:
7820:
7730:
7718:
7639:
7451:
7203:
7077:
7073:
7060:
7051:
7038:
7031:
7022:
6981:
6953:
6944:
6847:
6461:
6455:
6439:
6427:
6045:
6042:
6036:
6024:
5924:
5918:
5895:
5889:
5879:such that two function values
5802:
5776:
5725:{\displaystyle x\mapsto e^{x}}
5709:
5652:
5625:
5594:
5587:is continuous on the interval
5577:
5499:
5487:
5467:
5455:
5420:
5381:
5373:
5357:
5349:
5297:
5289:
5239:
5233:
5222:
5194:
5182:
5078:to be uniformly continuous so
4972:
4968:
4962:
4953:
4941:
4934:
4839:
4833:
4775:
4771:
4758:
4749:
4736:
4729:
4701:
4673:
4528:
4514:
4290:
4287:
4281:
4272:
4266:
4260:
4246:
4236:
4224:
4084:{\displaystyle \forall x\in I}
4012:
3980:
3976:
3970:
3961:
3955:
3948:
3926:
3912:
3891:
3888:
3882:
3873:
3867:
3861:
3831:
3819:
3779:
3776:
3770:
3761:
3755:
3749:
3735:
3725:
3713:
3534:
3515:
3495:
3476:
3450:
2995:
2983:
2934:
2931:
2925:
2916:
2910:
2904:
2874:
2862:
2853:
2841:
2765:
2753:
2655:
2652:
2646:
2637:
2631:
2625:
2611:
2601:
2589:
2432:
2420:
2369:
2366:
2360:
2351:
2345:
2339:
2303:
2291:
2093:{\displaystyle \forall y\in X}
2064:{\displaystyle \forall x\in X}
1961:
1958:
1952:
1943:
1937:
1931:
1917:
1907:
1895:
1737:
1706:
1702:
1696:
1687:
1681:
1674:
1669:
1658:
1644:
1381:
1369:
1275:
1263:
1212:
1209:
1203:
1194:
1188:
1182:
1146:
1134:
985:
966:
946:
927:
901:
832:
826:
803:
791:
756:
750:
526:
522:
516:
507:
501:
494:
402:of real numbers is said to be
344:
338:
311:
305:
121:
115:
1:
9059:Fitzpatrick, Patrick (2006).
8990:
8846:{\displaystyle (x_{1},x_{2})}
7307:, the answer is given by the
7187:a complete metric space, and
6742:for any real-valued function
6295:{\displaystyle 2\varepsilon }
6208:{\displaystyle 2\varepsilon }
6074:{\displaystyle 2\varepsilon }
5438:{\displaystyle x\mapsto ax+b}
5252:exists (and is finite), then
5098:is not uniformly continuous.
5058:to satisfy the condition for
4487:
3371:is continuous on an interval
8305:, and the converse holds if
7869:). One would like to extend
6555:{\displaystyle \mathbb {R} }
6410:precisely if the difference
6272:{\displaystyle \delta >0}
6162:{\displaystyle \delta >0}
5950:{\displaystyle \varepsilon }
5872:{\displaystyle \delta >0}
5811:{\displaystyle x\to \infty }
5325:{\displaystyle \mathbb {R} }
5031:{\displaystyle \varepsilon }
4590:{\displaystyle \varepsilon }
4566:{\displaystyle \mathbb {R} }
4048:{\displaystyle \mathbb {R} }
3588:{\displaystyle I\subseteq X}
3154:{\displaystyle \varepsilon }
3114:{\displaystyle \varepsilon }
3094:{\displaystyle \varepsilon }
2710:{\displaystyle \varepsilon }
2517:is said to be continuous if
2241:{\displaystyle \delta >0}
1597:{\displaystyle \delta >0}
1078:{\displaystyle \delta >0}
866:(distance-preserving map).
706:{\displaystyle \varepsilon }
666:{\displaystyle \varepsilon }
228:{\displaystyle \varepsilon }
7:
9104:Encyclopedia of Mathematics
9097:Kudryavtsev, L.D. (2001) ,
8971:
8347:In the special case of two
5669:{\displaystyle x\to \pi /2}
5402:
5200:{\displaystyle [0,\infty )}
3987:{\displaystyle |f(x)-f(y)|}
3411:is uniformly continuous on
2222:there exists a real number
1747:{\displaystyle A\implies B}
1578:there exists a real number
1059:there exists a real number
10:
9206:
8763:there exists an entourage
7867:Intermediate Value Theorem
6782:). More specifically, let
6474:is infinitesimal whenever
6331:
5937:have the maximum distance
5155:If a real-valued function
4992:for positive real numbers
3837:{\displaystyle d_{1}(x,y)}
2677:Alternatively, a function
851:{\displaystyle f(x)=x^{2}}
105:, moves over the graph of
8349:topological vector spaces
6362:, a real-valued function
3540:{\displaystyle (Y,d_{2})}
3501:{\displaystyle (X,d_{1})}
3434:Continuity of a function
3141:that depends on the both
2497:Equivalently, a function
991:{\displaystyle (Y,d_{2})}
952:{\displaystyle (X,d_{1})}
646:{\displaystyle \epsilon }
459:{\displaystyle \epsilon }
9159:10.1016/j.hm.2004.11.003
9078:Kelley, John L. (1955).
8705:{\displaystyle f:X\to Y}
8645:{\displaystyle f:V\to W}
8415:{\displaystyle f:V\to W}
7471:, i.e., the image under
7309:Tietze extension theorem
6318:{\displaystyle 2\delta }
6185:{\displaystyle 2\delta }
6097:{\displaystyle 2\delta }
6051:{\displaystyle (x,f(x))}
5571:uniformly equicontinuous
5566:is uniformly continuous.
5398:Examples and nonexamples
3459:{\displaystyle f:X\to Y}
1629:{\displaystyle x,y\in X}
1110:{\displaystyle x,y\in X}
910:{\displaystyle f:X\to Y}
8965:compact Hausdorff space
8111:{\displaystyle Q\cap I}
6518:is continuous on a set
6496:is infinitesimal. Thus
6488:{\displaystyle \delta }
6349:Other characterizations
6010:{\displaystyle \delta }
5531:absolute value function
5144:closed bounded interval
5138:is uniformly continuous
5051:{\displaystyle \delta }
4919:{\displaystyle \delta }
4610:{\displaystyle \delta }
4575:the set of real numbers
4415:{\displaystyle \delta }
4350:{\displaystyle \delta }
4032:the set of real numbers
3225:of the function domain
3134:{\displaystyle \delta }
3101:) that depends on only
3034:{\displaystyle \delta }
686:{\displaystyle \delta }
653:) that depends on only
626:{\displaystyle \delta }
599:{\displaystyle \delta }
479:{\displaystyle \delta }
439:{\displaystyle \delta }
419:{\displaystyle \delta }
268:{\displaystyle \delta }
255:-interval smaller than
8951:
8931:
8867:
8847:
8800:
8778:
8756:
8734:
8706:
8646:
8619:linear transformations
8609:
8542:
8496:
8476:
8456:
8436:
8416:
8384:
8364:
8336:Fourier transformation
8320:
8298:
8276:
8255:
8219:
8197:
8175:
8153:
8132:
8112:
8085:
8063:
8041:
8019:
7995:
7903:
7883:
7858:
7837:
7798:
7797:{\displaystyle a>1}
7768:
7747:
7695:
7677:is Cauchy-continuous.
7670:
7649:
7616:
7594:
7572:
7550:
7528:
7507:
7486:
7461:
7429:
7409:
7388:
7367:
7347:
7326:
7301:
7280:
7255:
7235:
7213:
7181:
7159:
7139:
7119:
7091:
6995:
6920:
6893:
6866:
6825:
6796:
6756:
6736:
6709:
6678:
6649:
6629:
6609:
6608:{\displaystyle a\in A}
6583:
6556:
6533:
6511:
6489:
6468:
6403:
6384:of a real variable is
6377:
6319:
6296:
6273:
6247:
6209:
6186:
6163:
6137:
6098:
6075:
6052:
6011:
5991:
5971:
5951:
5931:
5902:
5873:
5847:
5812:
5786:
5753:
5726:
5693:
5670:
5632:
5553:
5506:
5474:
5439:
5388:
5326:
5304:
5266:
5246:
5201:
5169:
5092:
5072:
5052:
5032:
5012:
4986:
4920:
4900:
4877:
4789:
4715:
4659:
4611:
4591:
4567:
4545:
4478:
4436:
4416:
4393:
4351:
4327:
4303:
4143:
4114:
4085:
4049:
4024:
3988:
3934:
3898:
3838:
3792:
3629:
3609:
3589:
3562:
3541:
3502:
3460:
3425:
3405:
3391:and a definition that
3385:
3365:
3338:
3318:
3287:
3267:
3239:
3219:
3199:
3175:
3155:
3135:
3115:
3095:
3075:
3055:
3035:
3002:
2967:
2947:
2881:
2818:
2817:{\displaystyle y\in X}
2792:
2772:
2737:
2736:{\displaystyle x\in X}
2711:
2691:
2668:
2511:
2488:
2468:
2454:is a neighbourhood of
2448:
2382:
2316:
2268:
2267:{\displaystyle y\in X}
2242:
2216:
2187:
2150:
2117:
2094:
2065:
2036:
2007:
1974:
1817:
1792:
1772:
1748:
1720:
1630:
1598:
1572:
1542:
1515:
1484:
1464:
1437:
1423:is a neighbourhood of
1417:
1397:
1331:
1317:is a neighbourhood of
1311:
1291:
1225:
1159:
1111:
1079:
1053:
1020:
992:
953:
911:
852:
810:
778:
727:
707:
687:
667:
647:
627:
600:
580:
560:
540:
480:
460:
440:
420:
396:
372:
361:
318:
289:
269:
249:
229:
209:
189:
169:
143:
99:
58:
9185:Mathematical analysis
8952:
8932:
8868:
8848:
8801:
8779:
8757:
8735:
8707:
8647:
8610:
8543:
8497:
8477:
8457:
8437:
8417:
8385:
8365:
8321:
8299:
8277:
8256:
8220:
8198:
8176:
8154:
8133:
8113:
8086:
8064:
8042:
8020:
7996:
7904:
7884:
7859:
7838:
7799:
7778:For example, suppose
7769:
7748:
7696:
7671:
7650:
7617:
7595:
7573:
7551:
7529:
7508:
7487:
7462:
7430:
7410:
7389:
7368:
7348:
7327:
7302:
7281:
7256:
7236:
7214:
7182:
7160:
7140:
7120:
7092:
6996:
6921:
6919:{\displaystyle y_{n}}
6894:
6892:{\displaystyle x_{n}}
6867:
6826:
6797:
6764:non-standard calculus
6757:
6737:
6735:{\displaystyle f^{*}}
6710:
6679:
6677:{\displaystyle ^{*}A}
6650:
6630:
6610:
6584:
6582:{\displaystyle f^{*}}
6557:
6534:
6512:
6490:
6469:
6404:
6378:
6360:non-standard analysis
6354:Non-standard analysis
6320:
6297:
6274:
6248:
6210:
6187:
6164:
6138:
6099:
6076:
6053:
6017:. Thus at each point
6012:
5992:
5972:
5952:
5932:
5903:
5874:
5848:
5813:
5787:
5754:
5752:{\displaystyle e^{x}}
5727:
5694:
5671:
5633:
5554:
5507:
5475:
5440:
5389:
5327:
5305:
5267:
5247:
5202:
5170:
5150:Darboux integrability
5103:absolutely continuous
5093:
5073:
5053:
5033:
5013:
4987:
4921:
4901:
4878:
4790:
4716:
4660:
4612:
4592:
4568:
4546:
4479:
4437:
4417:
4394:
4352:
4328:
4304:
4144:
4115:
4086:
4050:
4025:
3989:
3935:
3933:{\displaystyle |x-y|}
3899:
3839:
3793:
3630:
3610:
3595:(i.e., continuity of
3590:
3563:
3542:
3503:
3461:
3426:
3406:
3386:
3366:
3339:
3319:
3317:{\displaystyle f^{*}}
3288:
3268:
3240:
3220:
3200:
3176:
3156:
3136:
3116:
3096:
3076:
3056:
3036:
3003:
2968:
2948:
2882:
2819:
2793:
2773:
2738:
2712:
2692:
2669:
2512:
2489:
2469:
2449:
2383:
2317:
2269:
2243:
2217:
2188:
2151:
2125:modulus of continuity
2118:
2095:
2066:
2037:
2008:
1975:
1818:
1793:
1773:
1758:statement saying "if
1749:
1721:
1631:
1599:
1573:
1543:
1541:{\displaystyle d_{2}}
1516:
1514:{\displaystyle d_{1}}
1485:
1465:
1438:
1418:
1398:
1332:
1312:
1292:
1226:
1160:
1112:
1080:
1054:
1021:
993:
954:
912:
853:
811:
809:{\displaystyle (0,1)}
779:
728:
708:
693:that depends on both
688:
668:
648:
628:
601:
581:
561:
541:
481:
461:
441:
421:
397:
371:uniformly continuous.
367:, on the other hand,
362:
319:
290:
270:
250:
230:
210:
190:
170:
144:
100:
59:
22:
9147:Historia Mathematica
9099:"Uniform continuity"
8941:
8877:
8857:
8811:
8807:such that for every
8790:
8768:
8746:
8724:
8714:uniformly continuous
8684:
8624:
8552:
8506:
8486:
8466:
8446:
8426:
8394:
8374:
8354:
8310:
8288:
8266:
8233:
8209:
8187:
8165:
8143:
8122:
8096:
8075:
8053:
8031:
8009:
7916:
7893:
7873:
7848:
7808:
7782:
7758:
7706:
7685:
7660:
7627:
7606:
7584:
7562:
7540:
7518:
7497:
7476:
7439:
7419:
7399:
7378:
7357:
7337:
7316:
7291:
7270:
7245:
7225:
7191:
7171:
7149:
7129:
7109:
7011:
6933:
6903:
6876:
6835:
6806:
6786:
6746:
6719:
6688:
6659:
6639:
6619:
6593:
6566:
6544:
6523:
6501:
6479:
6414:
6393:
6367:
6306:
6283:
6257:
6231:
6196:
6173:
6147:
6121:
6085:
6062:
6021:
6001:
5981:
5961:
5941:
5930:{\displaystyle f(y)}
5912:
5901:{\displaystyle f(x)}
5883:
5857:
5831:
5796:
5763:
5736:
5703:
5683:
5646:
5591:
5564:Weierstrass function
5537:
5520:Lipschitz continuous
5484:
5452:
5414:
5336:
5314:
5276:
5256:
5211:
5179:
5159:
5130:Heine–Cantor theorem
5082:
5062:
5042:
5022:
4996:
4930:
4910:
4890:
4802:
4725:
4669:
4621:
4601:
4581:
4555:
4500:
4449:
4426:
4406:
4364:
4341:
4317:
4156:
4124:
4095:
4066:
4037:
3998:
3944:
3908:
3848:
3806:
3645:
3619:
3599:
3573:
3552:
3512:
3473:
3438:
3415:
3395:
3375:
3355:
3328:
3301:
3277:
3257:
3229:
3209:
3189:
3165:
3145:
3125:
3105:
3085:
3065:
3045:
3025:
2977:
2957:
2891:
2828:
2802:
2782:
2747:
2721:
2701:
2681:
2521:
2501:
2478:
2458:
2392:
2326:
2278:
2252:
2248:such that for every
2226:
2200:
2163:
2140:
2107:
2075:
2046:
2017:
1988:
1827:
1807:
1782:
1762:
1756:material conditional
1730:
1640:
1608:
1604:such that for every
1582:
1556:
1525:
1498:
1474:
1454:
1427:
1407:
1341:
1321:
1301:
1235:
1169:
1121:
1089:
1085:such that for every
1063:
1037:
1028:uniformly continuous
1010:
963:
924:
889:
820:
788:
744:
717:
697:
677:
657:
637:
617:
590:
570:
550:
490:
470:
450:
430:
410:
404:uniformly continuous
386:
332:
317:{\displaystyle f(x)}
299:
279:
259:
239:
219:
199:
179:
153:
149:in the direction of
109:
68:
27:
8984:Uniform isomorphism
8978:Contraction mapping
8676:continuity are the
8654:functional analysis
8070:the restriction of
7125:be a metric space,
5552:{\displaystyle x=0}
1490:are subsets of the
168:{\displaystyle x=0}
9029:Topologie Générale
8947:
8927:
8863:
8843:
8796:
8774:
8752:
8730:
8702:
8670:topological spaces
8642:
8605:
8538:
8492:
8472:
8452:
8432:
8412:
8380:
8360:
8316:
8294:
8272:
8251:
8215:
8193:
8171:
8149:
8128:
8108:
8081:
8059:
8037:
8015:
7991:
7899:
7879:
7854:
7833:
7794:
7764:
7743:
7691:
7666:
7645:
7612:
7590:
7568:
7557:to a metric space
7546:
7524:
7503:
7482:
7457:
7425:
7405:
7384:
7363:
7343:
7333:to the closure of
7322:
7297:
7276:
7251:
7231:
7209:
7177:
7155:
7135:
7115:
7087:
7029:
6991:
6951:
6916:
6889:
6862:
6821:
6792:
6752:
6732:
6705:
6674:
6645:
6625:
6605:
6579:
6552:
6529:
6507:
6485:
6464:
6399:
6373:
6315:
6292:
6269:
6243:
6205:
6182:
6159:
6133:
6094:
6071:
6048:
6007:
5987:
5967:
5947:
5927:
5898:
5869:
5843:
5808:
5782:
5749:
5722:
5689:
5666:
5628:
5569:Every member of a
5549:
5502:
5470:
5435:
5384:
5322:
5300:
5262:
5242:
5229:
5197:
5165:
5088:
5068:
5048:
5028:
5008:
4982:
4916:
4896:
4873:
4785:
4711:
4655:
4617:such that for all
4607:
4587:
4563:
4541:
4474:
4432:
4412:
4389:
4347:
4323:
4299:
4139:
4110:
4081:
4045:
4020:
3984:
3930:
3894:
3834:
3788:
3625:
3605:
3585:
3558:
3537:
3498:
3456:
3421:
3401:
3381:
3361:
3334:
3314:
3283:
3263:
3235:
3215:
3195:
3181:. Continuity is a
3171:
3151:
3131:
3111:
3091:
3071:
3051:
3031:
2998:
2963:
2943:
2877:
2814:
2788:
2768:
2733:
2707:
2687:
2664:
2507:
2484:
2464:
2444:
2378:
2312:
2264:
2238:
2212:
2183:
2181:
2146:
2113:
2090:
2061:
2032:
2003:
1970:
1813:
1788:
1768:
1744:
1716:
1626:
1594:
1568:
1538:
1511:
1480:
1460:
1433:
1413:
1393:
1327:
1307:
1287:
1221:
1155:
1107:
1075:
1049:
1016:
988:
949:
907:
848:
806:
774:
772:
723:
703:
683:
663:
643:
623:
596:
576:
556:
536:
476:
456:
436:
416:
392:
373:
357:
314:
285:
265:
245:
225:
205:
185:
165:
139:
137:
95:
54:
9137:978-0-07-054235-8
9061:Advanced Calculus
9054:. Academic Press.
9031:]. Springer.
9021:Bourbaki, Nicolas
8950:{\displaystyle V}
8866:{\displaystyle U}
8799:{\displaystyle X}
8777:{\displaystyle U}
8755:{\displaystyle Y}
8733:{\displaystyle V}
8495:{\displaystyle V}
8475:{\displaystyle A}
8455:{\displaystyle W}
8435:{\displaystyle B}
8383:{\displaystyle W}
8363:{\displaystyle V}
8319:{\displaystyle X}
8297:{\displaystyle X}
8275:{\displaystyle S}
8218:{\displaystyle R}
8196:{\displaystyle f}
8174:{\displaystyle I}
8152:{\displaystyle I}
8131:{\displaystyle f}
8084:{\displaystyle f}
8062:{\displaystyle I}
8040:{\displaystyle Q}
8018:{\displaystyle f}
7902:{\displaystyle R}
7882:{\displaystyle f}
7857:{\displaystyle x}
7767:{\displaystyle R}
7694:{\displaystyle X}
7669:{\displaystyle f}
7615:{\displaystyle f}
7593:{\displaystyle X}
7571:{\displaystyle Y}
7549:{\displaystyle X}
7527:{\displaystyle S}
7506:{\displaystyle X}
7485:{\displaystyle f}
7469:Cauchy-continuous
7428:{\displaystyle f}
7408:{\displaystyle X}
7387:{\displaystyle S}
7366:{\displaystyle X}
7346:{\displaystyle S}
7325:{\displaystyle f}
7300:{\displaystyle X}
7279:{\displaystyle S}
7254:{\displaystyle X}
7234:{\displaystyle f}
7180:{\displaystyle R}
7158:{\displaystyle X}
7138:{\displaystyle S}
7118:{\displaystyle X}
7014:
6936:
6795:{\displaystyle A}
6776:Cauchy continuity
6770:Cauchy continuity
6755:{\displaystyle f}
6648:{\displaystyle A}
6628:{\displaystyle f}
6532:{\displaystyle A}
6510:{\displaystyle f}
6402:{\displaystyle a}
6376:{\displaystyle f}
5990:{\displaystyle y}
5970:{\displaystyle x}
5692:{\displaystyle x}
5524:Hölder continuous
5512:is a compact set.
5265:{\displaystyle f}
5214:
5175:is continuous on
5168:{\displaystyle f}
5140:. In particular,
5091:{\displaystyle f}
5071:{\displaystyle f}
4899:{\displaystyle x}
4442:of the interval,
4435:{\displaystyle x}
4326:{\displaystyle x}
3628:{\displaystyle I}
3608:{\displaystyle f}
3561:{\displaystyle x}
3424:{\displaystyle I}
3404:{\displaystyle f}
3384:{\displaystyle I}
3364:{\displaystyle f}
3337:{\displaystyle f}
3286:{\displaystyle X}
3266:{\displaystyle f}
3238:{\displaystyle X}
3218:{\displaystyle x}
3198:{\displaystyle f}
3174:{\displaystyle x}
3074:{\displaystyle Y}
3054:{\displaystyle X}
2966:{\displaystyle x}
2791:{\displaystyle x}
2690:{\displaystyle f}
2510:{\displaystyle f}
2487:{\displaystyle x}
2467:{\displaystyle x}
2172:
2167:
2149:{\displaystyle f}
2116:{\displaystyle f}
1816:{\displaystyle f}
1791:{\displaystyle B}
1771:{\displaystyle A}
1483:{\displaystyle Y}
1463:{\displaystyle X}
1436:{\displaystyle y}
1416:{\displaystyle y}
1330:{\displaystyle x}
1310:{\displaystyle x}
1019:{\displaystyle f}
771:
726:{\displaystyle x}
579:{\displaystyle y}
559:{\displaystyle x}
395:{\displaystyle f}
355:
288:{\displaystyle f}
248:{\displaystyle x}
208:{\displaystyle f}
188:{\displaystyle f}
136:
9197:
9190:General topology
9161:
9141:
9125:
9111:
9093:
9080:General topology
9074:
9055:
9042:
9007:
9001:
8956:
8954:
8953:
8948:
8936:
8934:
8933:
8928:
8920:
8919:
8898:
8897:
8872:
8870:
8869:
8864:
8852:
8850:
8849:
8844:
8839:
8838:
8826:
8825:
8805:
8803:
8802:
8797:
8783:
8781:
8780:
8775:
8761:
8759:
8758:
8753:
8739:
8737:
8736:
8731:
8711:
8709:
8708:
8703:
8651:
8649:
8648:
8643:
8614:
8612:
8611:
8606:
8592:
8591:
8570:
8569:
8547:
8545:
8544:
8539:
8531:
8530:
8518:
8517:
8501:
8499:
8498:
8493:
8481:
8479:
8478:
8473:
8461:
8459:
8458:
8453:
8441:
8439:
8438:
8433:
8421:
8419:
8418:
8413:
8389:
8387:
8386:
8381:
8369:
8367:
8366:
8361:
8325:
8323:
8322:
8317:
8303:
8301:
8300:
8295:
8281:
8279:
8278:
8273:
8260:
8258:
8257:
8252:
8224:
8222:
8221:
8216:
8202:
8200:
8199:
8194:
8180:
8178:
8177:
8172:
8158:
8156:
8155:
8150:
8137:
8135:
8134:
8129:
8117:
8115:
8114:
8109:
8090:
8088:
8087:
8082:
8068:
8066:
8065:
8060:
8046:
8044:
8043:
8038:
8024:
8022:
8021:
8016:
8000:
7998:
7997:
7992:
7990:
7986:
7979:
7978:
7964:
7963:
7908:
7906:
7905:
7900:
7888:
7886:
7885:
7880:
7863:
7861:
7860:
7855:
7842:
7840:
7839:
7834:
7832:
7831:
7803:
7801:
7800:
7795:
7773:
7771:
7770:
7765:
7752:
7750:
7749:
7744:
7742:
7741:
7700:
7698:
7697:
7692:
7675:
7673:
7672:
7667:
7654:
7652:
7651:
7646:
7621:
7619:
7618:
7613:
7599:
7597:
7596:
7591:
7577:
7575:
7574:
7569:
7555:
7553:
7552:
7547:
7533:
7531:
7530:
7525:
7512:
7510:
7509:
7504:
7491:
7489:
7488:
7483:
7466:
7464:
7463:
7458:
7434:
7432:
7431:
7426:
7414:
7412:
7411:
7406:
7393:
7391:
7390:
7385:
7372:
7370:
7369:
7364:
7352:
7350:
7349:
7344:
7331:
7329:
7328:
7323:
7306:
7304:
7303:
7298:
7285:
7283:
7282:
7277:
7260:
7258:
7257:
7252:
7240:
7238:
7237:
7232:
7218:
7216:
7215:
7210:
7186:
7184:
7183:
7178:
7164:
7162:
7161:
7156:
7144:
7142:
7141:
7136:
7124:
7122:
7121:
7116:
7096:
7094:
7093:
7088:
7080:
7072:
7071:
7050:
7049:
7034:
7028:
7000:
6998:
6997:
6992:
6984:
6979:
6978:
6966:
6965:
6956:
6950:
6925:
6923:
6922:
6917:
6915:
6914:
6898:
6896:
6895:
6890:
6888:
6887:
6871:
6869:
6868:
6863:
6861:
6860:
6855:
6831:. If a function
6830:
6828:
6827:
6822:
6820:
6819:
6814:
6801:
6799:
6798:
6793:
6780:Fitzpatrick 2006
6761:
6759:
6758:
6753:
6741:
6739:
6738:
6733:
6731:
6730:
6714:
6712:
6711:
6706:
6704:
6699:
6698:
6683:
6681:
6680:
6675:
6670:
6669:
6654:
6652:
6651:
6646:
6634:
6632:
6631:
6626:
6614:
6612:
6611:
6606:
6588:
6586:
6585:
6580:
6578:
6577:
6561:
6559:
6558:
6553:
6551:
6538:
6536:
6535:
6530:
6516:
6514:
6513:
6508:
6494:
6492:
6491:
6486:
6473:
6471:
6470:
6465:
6454:
6453:
6426:
6425:
6408:
6406:
6405:
6400:
6382:
6380:
6379:
6374:
6324:
6322:
6321:
6316:
6301:
6299:
6298:
6293:
6278:
6276:
6275:
6270:
6252:
6250:
6249:
6244:
6224:
6214:
6212:
6211:
6206:
6191:
6189:
6188:
6183:
6168:
6166:
6165:
6160:
6142:
6140:
6139:
6134:
6114:
6103:
6101:
6100:
6095:
6080:
6078:
6077:
6072:
6057:
6055:
6054:
6049:
6016:
6014:
6013:
6008:
5996:
5994:
5993:
5988:
5976:
5974:
5973:
5968:
5956:
5954:
5953:
5948:
5936:
5934:
5933:
5928:
5907:
5905:
5904:
5899:
5878:
5876:
5875:
5870:
5852:
5850:
5849:
5844:
5817:
5815:
5814:
5809:
5791:
5789:
5788:
5783:
5775:
5774:
5758:
5756:
5755:
5750:
5748:
5747:
5731:
5729:
5728:
5723:
5721:
5720:
5698:
5696:
5695:
5690:
5675:
5673:
5672:
5667:
5662:
5637:
5635:
5634:
5629:
5621:
5607:
5585:tangent function
5558:
5556:
5555:
5550:
5511:
5509:
5508:
5505:{\displaystyle }
5503:
5479:
5477:
5476:
5473:{\displaystyle }
5471:
5444:
5442:
5441:
5436:
5409:Linear functions
5393:
5391:
5390:
5385:
5380:
5372:
5371:
5356:
5348:
5347:
5331:
5329:
5328:
5323:
5321:
5309:
5307:
5306:
5301:
5296:
5288:
5287:
5271:
5269:
5268:
5263:
5251:
5249:
5248:
5243:
5228:
5206:
5204:
5203:
5198:
5174:
5172:
5171:
5166:
5123:Euclidean metric
5097:
5095:
5094:
5089:
5077:
5075:
5074:
5069:
5057:
5055:
5054:
5049:
5037:
5035:
5034:
5029:
5017:
5015:
5014:
5009:
4991:
4989:
4988:
4983:
4975:
4937:
4925:
4923:
4922:
4917:
4905:
4903:
4902:
4897:
4882:
4880:
4879:
4874:
4869:
4868:
4826:
4822:
4794:
4792:
4791:
4786:
4778:
4770:
4769:
4748:
4747:
4732:
4720:
4718:
4717:
4712:
4704:
4699:
4698:
4686:
4685:
4676:
4664:
4662:
4661:
4656:
4654:
4646:
4645:
4633:
4632:
4616:
4614:
4613:
4608:
4596:
4594:
4593:
4588:
4572:
4570:
4569:
4564:
4562:
4550:
4548:
4547:
4542:
4540:
4539:
4521:
4513:
4483:
4481:
4480:
4475:
4441:
4439:
4438:
4433:
4421:
4419:
4418:
4413:
4398:
4396:
4395:
4390:
4356:
4354:
4353:
4348:
4332:
4330:
4329:
4324:
4308:
4306:
4305:
4300:
4259:
4258:
4223:
4222:
4148:
4146:
4145:
4140:
4119:
4117:
4116:
4111:
4090:
4088:
4087:
4082:
4054:
4052:
4051:
4046:
4044:
4029:
4027:
4026:
4021:
4019:
4011:
3993:
3991:
3990:
3985:
3983:
3951:
3939:
3937:
3936:
3931:
3929:
3915:
3903:
3901:
3900:
3895:
3860:
3859:
3843:
3841:
3840:
3835:
3818:
3817:
3797:
3795:
3794:
3789:
3748:
3747:
3712:
3711:
3634:
3632:
3631:
3626:
3615:on the interval
3614:
3612:
3611:
3606:
3594:
3592:
3591:
3586:
3567:
3565:
3564:
3559:
3546:
3544:
3543:
3538:
3533:
3532:
3507:
3505:
3504:
3499:
3494:
3493:
3465:
3463:
3462:
3457:
3430:
3428:
3427:
3422:
3410:
3408:
3407:
3402:
3390:
3388:
3387:
3382:
3370:
3368:
3367:
3362:
3343:
3341:
3340:
3335:
3323:
3321:
3320:
3315:
3313:
3312:
3292:
3290:
3289:
3284:
3272:
3270:
3269:
3264:
3244:
3242:
3241:
3236:
3224:
3222:
3221:
3216:
3204:
3202:
3201:
3196:
3180:
3178:
3177:
3172:
3160:
3158:
3157:
3152:
3140:
3138:
3137:
3132:
3120:
3118:
3117:
3112:
3100:
3098:
3097:
3092:
3080:
3078:
3077:
3072:
3060:
3058:
3057:
3052:
3040:
3038:
3037:
3032:
3007:
3005:
3004:
2999:
2972:
2970:
2969:
2964:
2952:
2950:
2949:
2944:
2903:
2902:
2886:
2884:
2883:
2878:
2840:
2839:
2823:
2821:
2820:
2815:
2797:
2795:
2794:
2789:
2777:
2775:
2774:
2769:
2742:
2740:
2739:
2734:
2716:
2714:
2713:
2708:
2696:
2694:
2693:
2688:
2673:
2671:
2670:
2665:
2624:
2623:
2588:
2587:
2516:
2514:
2513:
2508:
2493:
2491:
2490:
2485:
2473:
2471:
2470:
2465:
2453:
2451:
2450:
2445:
2419:
2418:
2387:
2385:
2384:
2379:
2338:
2337:
2321:
2319:
2318:
2313:
2290:
2289:
2273:
2271:
2270:
2265:
2247:
2245:
2244:
2239:
2221:
2219:
2218:
2213:
2192:
2190:
2189:
2184:
2182:
2177:
2173:
2170:
2155:
2153:
2152:
2147:
2122:
2120:
2119:
2114:
2099:
2097:
2096:
2091:
2070:
2068:
2067:
2062:
2041:
2039:
2038:
2033:
2012:
2010:
2009:
2004:
1979:
1977:
1976:
1971:
1930:
1929:
1894:
1893:
1822:
1820:
1819:
1814:
1797:
1795:
1794:
1789:
1777:
1775:
1774:
1769:
1753:
1751:
1750:
1745:
1725:
1723:
1722:
1717:
1709:
1677:
1661:
1647:
1635:
1633:
1632:
1627:
1603:
1601:
1600:
1595:
1577:
1575:
1574:
1569:
1547:
1545:
1544:
1539:
1537:
1536:
1520:
1518:
1517:
1512:
1510:
1509:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1368:
1367:
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1262:
1261:
1230:
1228:
1227:
1222:
1181:
1180:
1164:
1162:
1161:
1156:
1133:
1132:
1116:
1114:
1113:
1108:
1084:
1082:
1081:
1076:
1058:
1056:
1055:
1050:
1025:
1023:
1022:
1017:
997:
995:
994:
989:
984:
983:
958:
956:
955:
950:
945:
944:
916:
914:
913:
908:
857:
855:
854:
849:
847:
846:
815:
813:
812:
807:
783:
781:
780:
775:
773:
764:
732:
730:
729:
724:
712:
710:
709:
704:
692:
690:
689:
684:
672:
670:
669:
664:
652:
650:
649:
644:
632:
630:
629:
624:
605:
603:
602:
597:
585:
583:
582:
577:
565:
563:
562:
557:
545:
543:
542:
537:
529:
497:
485:
483:
482:
477:
465:
463:
462:
457:
445:
443:
442:
437:
425:
423:
422:
417:
401:
399:
398:
393:
366:
364:
363:
358:
356:
351:
323:
321:
320:
315:
294:
292:
291:
286:
274:
272:
271:
266:
254:
252:
251:
246:
234:
232:
231:
226:
214:
212:
211:
206:
194:
192:
191:
186:
174:
172:
171:
166:
148:
146:
145:
140:
138:
129:
104:
102:
101:
96:
94:
93:
85:
63:
61:
60:
55:
53:
52:
44:
9205:
9204:
9200:
9199:
9198:
9196:
9195:
9194:
9165:
9164:
9138:
9090:
9071:
9063:. Brooks/Cole.
9048:Dieudonné, Jean
9039:
9016:
9014:Further reading
9011:
9010:
9002:
8998:
8993:
8974:
8942:
8939:
8938:
8915:
8911:
8893:
8889:
8878:
8875:
8874:
8858:
8855:
8854:
8834:
8830:
8821:
8817:
8812:
8809:
8808:
8791:
8788:
8787:
8769:
8766:
8765:
8747:
8744:
8743:
8725:
8722:
8721:
8685:
8682:
8681:
8666:
8625:
8622:
8621:
8587:
8583:
8565:
8561:
8553:
8550:
8549:
8526:
8522:
8513:
8509:
8507:
8504:
8503:
8487:
8484:
8483:
8467:
8464:
8463:
8447:
8444:
8443:
8427:
8424:
8423:
8395:
8392:
8391:
8375:
8372:
8371:
8355:
8352:
8351:
8345:
8329:locally compact
8311:
8308:
8307:
8289:
8286:
8285:
8267:
8264:
8263:
8234:
8231:
8230:
8210:
8207:
8206:
8188:
8185:
8184:
8166:
8163:
8162:
8144:
8141:
8140:
8123:
8120:
8119:
8097:
8094:
8093:
8076:
8073:
8072:
8054:
8051:
8050:
8032:
8029:
8028:
8010:
8007:
8006:
7974:
7970:
7969:
7965:
7959:
7955:
7917:
7914:
7913:
7909:. The identity
7894:
7891:
7890:
7874:
7871:
7870:
7849:
7846:
7845:
7827:
7823:
7809:
7806:
7805:
7783:
7780:
7779:
7759:
7756:
7755:
7737:
7733:
7707:
7704:
7703:
7686:
7683:
7682:
7661:
7658:
7657:
7655:if and only if
7628:
7625:
7624:
7607:
7604:
7603:
7585:
7582:
7581:
7563:
7560:
7559:
7541:
7538:
7537:
7519:
7516:
7515:
7498:
7495:
7494:
7477:
7474:
7473:
7440:
7437:
7436:
7420:
7417:
7416:
7400:
7397:
7396:
7379:
7376:
7375:
7358:
7355:
7354:
7338:
7335:
7334:
7317:
7314:
7313:
7292:
7289:
7288:
7271:
7268:
7267:
7246:
7243:
7242:
7226:
7223:
7222:
7192:
7189:
7188:
7172:
7169:
7168:
7150:
7147:
7146:
7130:
7127:
7126:
7110:
7107:
7106:
7103:
7076:
7067:
7063:
7045:
7041:
7030:
7018:
7012:
7009:
7008:
6980:
6974:
6970:
6961:
6957:
6952:
6940:
6934:
6931:
6930:
6910:
6906:
6904:
6901:
6900:
6883:
6879:
6877:
6874:
6873:
6856:
6851:
6850:
6836:
6833:
6832:
6815:
6810:
6809:
6807:
6804:
6803:
6802:be a subset of
6787:
6784:
6783:
6772:
6747:
6744:
6743:
6726:
6722:
6720:
6717:
6716:
6700:
6694:
6691:
6689:
6686:
6685:
6665:
6662:
6660:
6657:
6656:
6640:
6637:
6636:
6620:
6617:
6616:
6594:
6591:
6590:
6573:
6569:
6567:
6564:
6563:
6547:
6545:
6542:
6541:
6524:
6521:
6520:
6502:
6499:
6498:
6480:
6477:
6476:
6449:
6445:
6421:
6417:
6415:
6412:
6411:
6394:
6391:
6390:
6386:microcontinuous
6368:
6365:
6364:
6356:
6351:
6334:
6327:
6307:
6304:
6303:
6284:
6281:
6280:
6258:
6255:
6254:
6232:
6229:
6228:
6225:
6216:
6197:
6194:
6193:
6174:
6171:
6170:
6148:
6145:
6144:
6122:
6119:
6118:
6115:
6086:
6083:
6082:
6063:
6060:
6059:
6022:
6019:
6018:
6002:
5999:
5998:
5982:
5979:
5978:
5962:
5959:
5958:
5942:
5939:
5938:
5913:
5910:
5909:
5884:
5881:
5880:
5858:
5855:
5854:
5832:
5829:
5828:
5825:
5797:
5794:
5793:
5770:
5766:
5764:
5761:
5760:
5743:
5739:
5737:
5734:
5733:
5716:
5712:
5704:
5701:
5700:
5684:
5681:
5680:
5658:
5647:
5644:
5643:
5617:
5603:
5592:
5589:
5588:
5580:
5538:
5535:
5534:
5485:
5482:
5481:
5453:
5450:
5449:
5415:
5412:
5411:
5405:
5400:
5376:
5367:
5363:
5352:
5343:
5339:
5337:
5334:
5333:
5317:
5315:
5312:
5311:
5292:
5283:
5279:
5277:
5274:
5273:
5257:
5254:
5253:
5218:
5212:
5209:
5208:
5180:
5177:
5176:
5160:
5157:
5156:
5119:discrete metric
5114:totally bounded
5112:The image of a
5107:Cantor function
5083:
5080:
5079:
5063:
5060:
5059:
5043:
5040:
5039:
5023:
5020:
5019:
4997:
4994:
4993:
4971:
4933:
4931:
4928:
4927:
4911:
4908:
4907:
4891:
4888:
4887:
4864:
4860:
4812:
4808:
4803:
4800:
4799:
4774:
4765:
4761:
4743:
4739:
4728:
4726:
4723:
4722:
4700:
4694:
4690:
4681:
4677:
4672:
4670:
4667:
4666:
4650:
4641:
4637:
4628:
4624:
4622:
4619:
4618:
4602:
4599:
4598:
4582:
4579:
4578:
4558:
4556:
4553:
4552:
4535:
4531:
4517:
4509:
4501:
4498:
4497:
4490:
4450:
4447:
4446:
4427:
4424:
4423:
4407:
4404:
4403:
4365:
4362:
4361:
4342:
4339:
4338:
4333:of the interval
4318:
4315:
4314:
4254:
4250:
4218:
4214:
4157:
4154:
4153:
4149:) are rotated:
4125:
4122:
4121:
4096:
4093:
4092:
4067:
4064:
4063:
4060:quantifications
4040:
4038:
4035:
4034:
4015:
4007:
3999:
3996:
3995:
3979:
3947:
3945:
3942:
3941:
3925:
3911:
3909:
3906:
3905:
3855:
3851:
3849:
3846:
3845:
3813:
3809:
3807:
3804:
3803:
3743:
3739:
3707:
3703:
3646:
3643:
3642:
3637:quantifications
3620:
3617:
3616:
3600:
3597:
3596:
3574:
3571:
3570:
3569:of an interval
3553:
3550:
3549:
3547:at every point
3528:
3524:
3513:
3510:
3509:
3489:
3485:
3474:
3471:
3470:
3439:
3436:
3435:
3416:
3413:
3412:
3396:
3393:
3392:
3376:
3373:
3372:
3356:
3353:
3352:
3329:
3326:
3325:
3308:
3304:
3302:
3299:
3298:
3278:
3275:
3274:
3258:
3255:
3254:
3230:
3227:
3226:
3210:
3207:
3206:
3190:
3187:
3186:
3166:
3163:
3162:
3146:
3143:
3142:
3126:
3123:
3122:
3106:
3103:
3102:
3086:
3083:
3082:
3066:
3063:
3062:
3046:
3043:
3042:
3026:
3023:
3022:
3015:
2978:
2975:
2974:
2958:
2955:
2954:
2898:
2894:
2892:
2889:
2888:
2835:
2831:
2829:
2826:
2825:
2803:
2800:
2799:
2783:
2780:
2779:
2748:
2745:
2744:
2722:
2719:
2718:
2702:
2699:
2698:
2682:
2679:
2678:
2619:
2615:
2583:
2579:
2522:
2519:
2518:
2502:
2499:
2498:
2479:
2476:
2475:
2459:
2456:
2455:
2414:
2410:
2393:
2390:
2389:
2333:
2329:
2327:
2324:
2323:
2285:
2281:
2279:
2276:
2275:
2253:
2250:
2249:
2227:
2224:
2223:
2201:
2198:
2197:
2169:
2168:
2166:
2164:
2161:
2160:
2141:
2138:
2137:
2134:
2108:
2105:
2104:
2076:
2073:
2072:
2047:
2044:
2043:
2018:
2015:
2014:
1989:
1986:
1985:
1982:quantifications
1925:
1921:
1889:
1885:
1828:
1825:
1824:
1808:
1805:
1804:
1783:
1780:
1779:
1763:
1760:
1759:
1731:
1728:
1727:
1705:
1673:
1657:
1643:
1641:
1638:
1637:
1609:
1606:
1605:
1583:
1580:
1579:
1557:
1554:
1553:
1532:
1528:
1526:
1523:
1522:
1505:
1501:
1499:
1496:
1495:
1475:
1472:
1471:
1455:
1452:
1451:
1428:
1425:
1424:
1408:
1405:
1404:
1363:
1359:
1342:
1339:
1338:
1322:
1319:
1318:
1302:
1299:
1298:
1257:
1253:
1236:
1233:
1232:
1176:
1172:
1170:
1167:
1166:
1128:
1124:
1122:
1119:
1118:
1090:
1087:
1086:
1064:
1061:
1060:
1038:
1035:
1034:
1011:
1008:
1007:
1004:
979:
975:
964:
961:
960:
940:
936:
925:
922:
921:
890:
887:
886:
885:For a function
883:
842:
838:
821:
818:
817:
789:
786:
785:
762:
745:
742:
741:
718:
715:
714:
698:
695:
694:
678:
675:
674:
658:
655:
654:
638:
635:
634:
618:
615:
614:
591:
588:
587:
571:
568:
567:
551:
548:
547:
525:
493:
491:
488:
487:
471:
468:
467:
451:
448:
447:
431:
428:
427:
411:
408:
407:
387:
384:
383:
350:
333:
330:
329:
300:
297:
296:
280:
277:
276:
260:
257:
256:
240:
237:
236:
220:
217:
216:
200:
197:
196:
180:
177:
176:
154:
151:
150:
127:
110:
107:
106:
86:
81:
80:
69:
66:
65:
64:and real width
45:
40:
39:
28:
25:
24:
17:
12:
11:
5:
9203:
9193:
9192:
9187:
9182:
9177:
9163:
9162:
9153:(3): 303–311,
9142:
9136:
9112:
9094:
9088:
9075:
9069:
9056:
9044:
9037:
9015:
9012:
9009:
9008:
8995:
8994:
8992:
8989:
8988:
8987:
8981:
8973:
8970:
8946:
8926:
8923:
8918:
8914:
8910:
8907:
8904:
8901:
8896:
8892:
8888:
8885:
8882:
8862:
8842:
8837:
8833:
8829:
8824:
8820:
8816:
8795:
8773:
8751:
8729:
8701:
8698:
8695:
8692:
8689:
8678:uniform spaces
8665:
8662:
8641:
8638:
8635:
8632:
8629:
8604:
8601:
8598:
8595:
8590:
8586:
8582:
8579:
8576:
8573:
8568:
8564:
8560:
8557:
8537:
8534:
8529:
8525:
8521:
8516:
8512:
8491:
8471:
8451:
8431:
8411:
8408:
8405:
8402:
8399:
8379:
8359:
8344:
8341:
8315:
8293:
8271:
8250:
8247:
8244:
8241:
8238:
8214:
8192:
8170:
8148:
8127:
8107:
8104:
8101:
8080:
8058:
8036:
8014:
8002:
8001:
7989:
7985:
7982:
7977:
7973:
7968:
7962:
7958:
7954:
7951:
7948:
7945:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7921:
7898:
7878:
7853:
7830:
7826:
7822:
7819:
7816:
7813:
7793:
7790:
7787:
7763:
7740:
7736:
7732:
7729:
7726:
7723:
7720:
7717:
7714:
7711:
7690:
7665:
7644:
7641:
7638:
7635:
7632:
7611:
7589:
7567:
7545:
7523:
7502:
7481:
7467:is that it is
7456:
7453:
7450:
7447:
7444:
7424:
7404:
7383:
7362:
7342:
7321:
7296:
7275:
7250:
7230:
7208:
7205:
7202:
7199:
7196:
7176:
7154:
7134:
7114:
7102:
7099:
7098:
7097:
7086:
7083:
7079:
7075:
7070:
7066:
7062:
7059:
7056:
7053:
7048:
7044:
7040:
7037:
7033:
7027:
7024:
7021:
7017:
7002:
7001:
6990:
6987:
6983:
6977:
6973:
6969:
6964:
6960:
6955:
6949:
6946:
6943:
6939:
6913:
6909:
6886:
6882:
6859:
6854:
6849:
6846:
6843:
6840:
6818:
6813:
6791:
6771:
6768:
6751:
6729:
6725:
6703:
6697:
6693:
6673:
6668:
6664:
6644:
6624:
6604:
6601:
6598:
6576:
6572:
6550:
6528:
6506:
6484:
6463:
6460:
6457:
6452:
6448:
6444:
6441:
6438:
6435:
6432:
6429:
6424:
6420:
6398:
6372:
6355:
6352:
6350:
6347:
6333:
6330:
6329:
6328:
6314:
6311:
6291:
6288:
6268:
6265:
6262:
6242:
6239:
6236:
6226:
6219:
6217:
6204:
6201:
6181:
6178:
6158:
6155:
6152:
6132:
6129:
6126:
6116:
6109:
6093:
6090:
6070:
6067:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6006:
5986:
5966:
5946:
5926:
5923:
5920:
5917:
5897:
5894:
5891:
5888:
5868:
5865:
5862:
5842:
5839:
5836:
5824:
5821:
5820:
5819:
5807:
5804:
5801:
5781:
5778:
5773:
5769:
5746:
5742:
5719:
5715:
5711:
5708:
5688:
5677:
5665:
5661:
5657:
5654:
5651:
5627:
5624:
5620:
5616:
5613:
5610:
5606:
5602:
5599:
5596:
5579:
5576:
5575:
5574:
5567:
5560:
5548:
5545:
5542:
5527:
5516:
5513:
5501:
5498:
5495:
5492:
5489:
5469:
5466:
5463:
5460:
5457:
5446:
5434:
5431:
5428:
5425:
5422:
5419:
5404:
5401:
5399:
5396:
5383:
5379:
5375:
5370:
5366:
5362:
5359:
5355:
5351:
5346:
5342:
5320:
5299:
5295:
5291:
5286:
5282:
5261:
5241:
5238:
5235:
5232:
5227:
5224:
5221:
5217:
5196:
5193:
5190:
5187:
5184:
5164:
5087:
5067:
5047:
5027:
5018:and the given
5007:
5004:
5001:
4981:
4978:
4974:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4936:
4915:
4895:
4884:
4883:
4872:
4867:
4863:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4825:
4821:
4818:
4815:
4811:
4807:
4784:
4781:
4777:
4773:
4768:
4764:
4760:
4757:
4754:
4751:
4746:
4742:
4738:
4735:
4731:
4710:
4707:
4703:
4697:
4693:
4689:
4684:
4680:
4675:
4653:
4649:
4644:
4640:
4636:
4631:
4627:
4606:
4586:
4561:
4538:
4534:
4530:
4527:
4524:
4520:
4516:
4512:
4508:
4505:
4489:
4486:
4485:
4484:
4473:
4470:
4467:
4464:
4460:
4457:
4454:
4431:
4411:
4400:
4399:
4388:
4385:
4382:
4379:
4375:
4372:
4369:
4346:
4322:
4311:
4310:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4257:
4253:
4248:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4221:
4217:
4212:
4209:
4206:
4203:
4200:
4196:
4193:
4190:
4187:
4183:
4180:
4177:
4174:
4170:
4167:
4164:
4161:
4138:
4135:
4132:
4129:
4109:
4106:
4103:
4100:
4080:
4077:
4074:
4071:
4043:
4018:
4014:
4010:
4006:
4003:
3982:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3950:
3928:
3924:
3921:
3918:
3914:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3858:
3854:
3833:
3830:
3827:
3824:
3821:
3816:
3812:
3800:
3799:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3746:
3742:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3710:
3706:
3701:
3698:
3695:
3692:
3689:
3685:
3682:
3679:
3676:
3672:
3669:
3666:
3663:
3659:
3656:
3653:
3650:
3624:
3604:
3584:
3581:
3578:
3557:
3536:
3531:
3527:
3523:
3520:
3517:
3497:
3492:
3488:
3484:
3481:
3478:
3455:
3452:
3449:
3446:
3443:
3420:
3400:
3380:
3360:
3333:
3311:
3307:
3282:
3262:
3234:
3214:
3194:
3170:
3150:
3130:
3110:
3090:
3081:are less than
3070:
3050:
3030:
3014:
3011:
3010:
3009:
2997:
2994:
2991:
2988:
2985:
2982:
2962:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2901:
2897:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2838:
2834:
2813:
2810:
2807:
2787:
2767:
2764:
2761:
2758:
2755:
2752:
2732:
2729:
2726:
2706:
2686:
2675:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2622:
2618:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2586:
2582:
2577:
2574:
2571:
2568:
2565:
2561:
2558:
2555:
2552:
2548:
2545:
2542:
2539:
2535:
2532:
2529:
2526:
2506:
2495:
2483:
2463:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2417:
2413:
2409:
2406:
2403:
2400:
2397:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2336:
2332:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2288:
2284:
2263:
2260:
2257:
2237:
2234:
2231:
2211:
2208:
2205:
2180:
2176:
2145:
2133:
2130:
2129:
2128:
2112:
2103:Equivalently,
2101:
2089:
2086:
2083:
2080:
2060:
2057:
2054:
2051:
2031:
2028:
2025:
2022:
2002:
1999:
1996:
1993:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1928:
1924:
1919:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1892:
1888:
1883:
1880:
1877:
1874:
1871:
1867:
1864:
1861:
1858:
1854:
1851:
1848:
1845:
1841:
1838:
1835:
1832:
1812:
1803:Equivalently,
1801:
1800:
1799:
1787:
1767:
1743:
1739:
1735:
1715:
1712:
1708:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1676:
1671:
1667:
1664:
1660:
1656:
1653:
1650:
1646:
1625:
1622:
1619:
1616:
1613:
1593:
1590:
1587:
1567:
1564:
1561:
1535:
1531:
1508:
1504:
1479:
1459:
1432:
1412:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1366:
1362:
1358:
1355:
1352:
1349:
1346:
1326:
1306:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1260:
1256:
1252:
1249:
1246:
1243:
1240:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1179:
1175:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1131:
1127:
1106:
1103:
1100:
1097:
1094:
1074:
1071:
1068:
1048:
1045:
1042:
1015:
1003:
1000:
987:
982:
978:
974:
971:
968:
948:
943:
939:
935:
932:
929:
906:
903:
900:
897:
894:
882:
879:
871:neighbourhoods
845:
841:
837:
834:
831:
828:
825:
805:
802:
799:
796:
793:
770:
767:
761:
758:
755:
752:
749:
722:
702:
682:
662:
642:
622:
595:
575:
555:
535:
532:
528:
524:
521:
518:
515:
512:
509:
506:
503:
500:
496:
475:
455:
435:
415:
391:
354:
349:
346:
343:
340:
337:
313:
310:
307:
304:
284:
264:
244:
224:
204:
184:
164:
161:
158:
135:
132:
126:
123:
120:
117:
114:
92:
89:
84:
79:
76:
73:
51:
48:
43:
38:
35:
32:
15:
9:
6:
4:
3:
2:
9202:
9191:
9188:
9186:
9183:
9181:
9178:
9176:
9173:
9172:
9170:
9160:
9156:
9152:
9148:
9143:
9139:
9133:
9129:
9124:
9123:
9117:
9116:Rudin, Walter
9113:
9110:
9106:
9105:
9100:
9095:
9091:
9089:0-387-90125-6
9085:
9081:
9076:
9072:
9070:0-534-92612-6
9066:
9062:
9057:
9053:
9049:
9045:
9040:
9038:0-387-19374-X
9034:
9030:
9026:
9022:
9018:
9017:
9005:
9000:
8996:
8985:
8982:
8979:
8976:
8975:
8969:
8966:
8961:
8958:
8944:
8916:
8912:
8905:
8902:
8894:
8890:
8883:
8860:
8835:
8831:
8827:
8822:
8818:
8806:
8793:
8784:
8771:
8762:
8749:
8740:
8727:
8719:
8716:if for every
8715:
8699:
8693:
8690:
8687:
8680:. A function
8679:
8675:
8671:
8661:
8659:
8655:
8639:
8633:
8630:
8627:
8620:
8615:
8602:
8599:
8596:
8588:
8584:
8577:
8574:
8566:
8562:
8555:
8535:
8532:
8527:
8523:
8519:
8514:
8510:
8489:
8469:
8449:
8429:
8409:
8403:
8400:
8397:
8377:
8357:
8350:
8340:
8337:
8332:
8330:
8326:
8313:
8304:
8291:
8282:
8269:
8248:
8242:
8239:
8236:
8227:
8225:
8212:
8203:
8190:
8181:
8168:
8159:
8146:
8125:
8105:
8102:
8099:
8091:
8078:
8069:
8056:
8047:
8034:
8025:
8012:
7987:
7983:
7980:
7975:
7971:
7966:
7960:
7956:
7952:
7946:
7940:
7937:
7931:
7928:
7925:
7919:
7912:
7911:
7910:
7896:
7876:
7868:
7864:
7851:
7828:
7824:
7817:
7814:
7811:
7791:
7788:
7785:
7776:
7774:
7761:
7738:
7734:
7727:
7724:
7721:
7715:
7712:
7709:
7701:
7688:
7678:
7676:
7663:
7642:
7636:
7633:
7630:
7622:
7609:
7601:is complete,
7600:
7587:
7578:
7565:
7556:
7543:
7534:
7521:
7500:
7492:
7479:
7470:
7454:
7448:
7445:
7442:
7422:
7402:
7394:
7381:
7360:
7340:
7332:
7319:
7310:
7294:
7287:is closed in
7286:
7273:
7263:
7262:
7248:
7228:
7206:
7200:
7197:
7194:
7174:
7167:
7152:
7132:
7112:
7084:
7081:
7068:
7064:
7057:
7054:
7046:
7042:
7035:
7019:
7007:
7006:
7005:
6988:
6985:
6975:
6971:
6967:
6962:
6958:
6941:
6929:
6928:
6927:
6911:
6907:
6884:
6880:
6857:
6844:
6841:
6838:
6816:
6789:
6781:
6777:
6767:
6765:
6749:
6727:
6723:
6695:
6692:
6671:
6666:
6663:
6642:
6622:
6602:
6599:
6596:
6574:
6570:
6562:precisely if
6539:
6526:
6517:
6504:
6495:
6482:
6458:
6450:
6446:
6442:
6436:
6433:
6430:
6422:
6418:
6409:
6396:
6387:
6383:
6370:
6361:
6346:
6343:
6339:
6312:
6309:
6289:
6286:
6266:
6263:
6260:
6240:
6237:
6234:
6223:
6218:
6202:
6199:
6179:
6176:
6156:
6153:
6150:
6130:
6127:
6124:
6113:
6108:
6107:
6106:
6091:
6088:
6068:
6065:
6039:
6033:
6030:
6027:
6004:
5984:
5964:
5944:
5921:
5915:
5892:
5886:
5866:
5863:
5860:
5840:
5837:
5834:
5823:Visualization
5799:
5771:
5767:
5744:
5740:
5717:
5713:
5706:
5686:
5678:
5663:
5659:
5655:
5649:
5641:
5622:
5618:
5614:
5611:
5608:
5604:
5600:
5597:
5586:
5582:
5581:
5572:
5568:
5565:
5561:
5546:
5543:
5540:
5532:
5528:
5525:
5521:
5517:
5514:
5496:
5493:
5490:
5464:
5461:
5458:
5447:
5432:
5429:
5426:
5423:
5417:
5410:
5407:
5406:
5395:
5368:
5364:
5360:
5344:
5340:
5284:
5280:
5259:
5236:
5230:
5219:
5188:
5185:
5162:
5153:
5151:
5147:
5145:
5139:
5137:
5132:asserts that
5131:
5126:
5124:
5120:
5115:
5110:
5108:
5104:
5099:
5085:
5065:
5045:
5025:
5005:
5002:
4999:
4979:
4976:
4965:
4959:
4956:
4950:
4947:
4944:
4938:
4913:
4893:
4870:
4865:
4861:
4857:
4854:
4851:
4848:
4845:
4842:
4836:
4830:
4827:
4823:
4819:
4816:
4813:
4809:
4805:
4798:
4797:
4796:
4782:
4779:
4766:
4762:
4755:
4752:
4744:
4740:
4733:
4708:
4705:
4695:
4691:
4687:
4682:
4678:
4647:
4642:
4638:
4634:
4629:
4625:
4604:
4584:
4576:
4536:
4532:
4525:
4522:
4506:
4503:
4495:
4471:
4468:
4465:
4458:
4452:
4445:
4444:
4443:
4429:
4409:
4386:
4383:
4380:
4373:
4367:
4360:
4359:
4358:
4344:
4336:
4320:
4296:
4293:
4284:
4278:
4275:
4269:
4263:
4255:
4251:
4242:
4239:
4233:
4230:
4227:
4219:
4215:
4210:
4207:
4204:
4201:
4194:
4191:
4188:
4181:
4178:
4175:
4168:
4165:
4162:
4152:
4151:
4150:
4136:
4133:
4130:
4107:
4104:
4101:
4078:
4075:
4072:
4061:
4056:
4033:
4004:
4001:
3973:
3967:
3964:
3958:
3952:
3922:
3919:
3916:
3885:
3879:
3876:
3870:
3864:
3856:
3852:
3828:
3825:
3822:
3814:
3810:
3785:
3782:
3773:
3767:
3764:
3758:
3752:
3744:
3740:
3731:
3728:
3722:
3719:
3716:
3708:
3704:
3699:
3696:
3693:
3690:
3683:
3680:
3677:
3670:
3667:
3664:
3657:
3654:
3651:
3641:
3640:
3639:
3638:
3622:
3602:
3582:
3579:
3576:
3568:
3555:
3529:
3525:
3521:
3518:
3490:
3486:
3482:
3479:
3469:
3468:metric spaces
3453:
3447:
3444:
3441:
3432:
3418:
3398:
3378:
3358:
3349:
3347:
3331:
3309:
3305:
3296:
3280:
3260:
3252:
3248:
3232:
3212:
3192:
3184:
3168:
3148:
3128:
3108:
3088:
3068:
3048:
3028:
3020:
2992:
2989:
2986:
2980:
2960:
2940:
2937:
2928:
2922:
2919:
2913:
2907:
2899:
2895:
2871:
2868:
2865:
2859:
2856:
2850:
2847:
2844:
2836:
2832:
2811:
2808:
2805:
2785:
2762:
2759:
2756:
2750:
2730:
2727:
2724:
2704:
2684:
2676:
2661:
2658:
2649:
2643:
2640:
2634:
2628:
2620:
2616:
2607:
2604:
2598:
2595:
2592:
2584:
2580:
2575:
2572:
2569:
2566:
2559:
2556:
2553:
2546:
2543:
2540:
2533:
2530:
2527:
2504:
2496:
2481:
2461:
2438:
2435:
2429:
2426:
2423:
2415:
2411:
2407:
2404:
2401:
2398:
2375:
2372:
2363:
2357:
2354:
2348:
2342:
2334:
2330:
2309:
2306:
2300:
2297:
2294:
2286:
2282:
2261:
2258:
2255:
2235:
2232:
2229:
2209:
2206:
2203:
2196:
2193:if for every
2178:
2174:
2159:
2143:
2136:
2135:
2126:
2110:
2102:
2087:
2084:
2081:
2058:
2055:
2052:
2029:
2026:
2023:
2000:
1997:
1994:
1983:
1967:
1964:
1955:
1949:
1946:
1940:
1934:
1926:
1922:
1913:
1910:
1904:
1901:
1898:
1890:
1886:
1881:
1878:
1875:
1872:
1865:
1862:
1859:
1852:
1849:
1846:
1839:
1836:
1833:
1810:
1802:
1785:
1765:
1757:
1741:
1733:
1713:
1710:
1699:
1693:
1690:
1684:
1678:
1665:
1662:
1654:
1651:
1648:
1623:
1620:
1617:
1614:
1611:
1591:
1588:
1585:
1565:
1562:
1559:
1551:
1533:
1529:
1506:
1502:
1493:
1477:
1457:
1449:
1448:
1446:
1430:
1410:
1387:
1384:
1378:
1375:
1372:
1364:
1360:
1356:
1353:
1350:
1347:
1324:
1304:
1281:
1278:
1272:
1269:
1266:
1258:
1254:
1250:
1247:
1244:
1241:
1218:
1215:
1206:
1200:
1197:
1191:
1185:
1177:
1173:
1152:
1149:
1143:
1140:
1137:
1129:
1125:
1104:
1101:
1098:
1095:
1092:
1072:
1069:
1066:
1046:
1043:
1040:
1033:
1030:if for every
1029:
1013:
1006:
1005:
999:
980:
976:
972:
969:
941:
937:
933:
930:
920:
919:metric spaces
904:
898:
895:
892:
878:
876:
875:uniform space
872:
867:
865:
861:
860:Lipschitz map
843:
839:
835:
829:
823:
800:
797:
794:
768:
765:
759:
753:
747:
738:
736:
735:metric spaces
720:
700:
680:
660:
640:
620:
612:
607:
593:
573:
553:
533:
530:
519:
513:
510:
504:
498:
473:
453:
433:
413:
405:
389:
382:
378:
370:
352:
347:
341:
335:
327:
308:
302:
282:
262:
242:
222:
202:
182:
162:
159:
156:
133:
130:
124:
118:
112:
90:
87:
77:
74:
71:
49:
46:
36:
33:
30:
21:
9150:
9146:
9126:. New York:
9121:
9102:
9079:
9060:
9051:
9028:
9024:
8999:
8962:
8959:
8786:
8764:
8742:
8720:
8713:
8673:
8667:
8658:Banach space
8616:
8346:
8333:
8306:
8284:
8262:
8228:
8205:
8183:
8161:
8139:
8071:
8049:
8027:
8005:
8003:
7844:
7777:
7754:
7681:
7679:
7656:
7602:
7580:
7558:
7536:
7514:
7472:
7395:is dense in
7374:
7312:
7266:
7264:
7220:
7165:
7145:a subset of
7104:
7003:
6773:
6519:
6497:
6475:
6389:
6363:
6357:
6335:
5826:
5639:
5154:
5141:
5133:
5127:
5111:
5100:
4885:
4491:
4401:
4334:
4312:
4057:
3801:
3548:
3433:
3350:
3294:
3253:property of
3250:
3182:
3016:
2157:
1337:and the set
1027:
884:
868:
739:
608:
403:
374:
368:
325:
9128:McGraw-Hill
8482:of zero in
8442:of zero in
8004:shows that
6388:at a point
5578:Nonexamples
5136:compact set
2953:. At every
2195:real number
2100:) are used.
1548:can be the
1032:real number
377:mathematics
9169:Categories
8991:References
8502:such that
6926:such that
4721:, we have
4494:continuous
4488:Properties
3019:continuity
2824:satisfies
2388:. The set
2322:, we have
2158:continuous
2156:is called
1231:. The set
1165:, we have
1026:is called
611:continuity
486:such that
9109:EMS Press
8718:entourage
8697:→
8637:→
8597:∈
8575:−
8533:∈
8520:−
8407:→
8246:→
8103:∩
7981:−
7976:δ
7938:−
7932:δ
7821:↦
7731:↦
7719:→
7640:→
7452:→
7221:When can
7204:→
7055:−
7026:∞
7023:→
6968:−
6948:∞
6945:→
6848:→
6728:∗
6696:∗
6667:∗
6600:∈
6575:∗
6483:δ
6451:∗
6443:−
6437:δ
6423:∗
6342:Dirichlet
6313:δ
6290:ε
6261:δ
6235:ε
6203:ε
6180:δ
6151:δ
6125:ε
6092:δ
6069:ε
6005:δ
5957:whenever
5945:ε
5861:δ
5835:ε
5806:∞
5803:→
5780:∞
5777:→
5710:↦
5656:π
5653:→
5615:π
5601:π
5598:−
5421:↦
5361:⊂
5226:∞
5223:→
5192:∞
5046:δ
5026:ε
5006:δ
5000:β
4980:ε
4957:−
4951:β
4914:δ
4862:δ
4855:δ
4852:⋅
4828:−
4820:δ
4783:ε
4753:−
4709:δ
4688:−
4648:∈
4605:δ
4585:ε
4529:↦
4515:→
4507::
4469:⋯
4463:∀
4459:δ
4456:∃
4453:⋯
4410:δ
4384:⋯
4381:δ
4378:∃
4371:∀
4368:⋯
4345:δ
4297:ε
4247:⇒
4243:δ
4205:∈
4199:∀
4192:∈
4186:∀
4176:δ
4173:∃
4163:ε
4160:∀
4131:δ
4128:∃
4102:ε
4099:∀
4076:∈
4070:∀
4013:→
3965:−
3920:−
3802:(metrics
3786:ε
3736:⇒
3732:δ
3694:∈
3688:∀
3678:δ
3675:∃
3665:ε
3662:∀
3655:∈
3649:∀
3580:⊆
3451:→
3310:∗
3149:ε
3129:δ
3109:ε
3089:ε
3029:δ
2987:ε
2981:δ
2941:ε
2866:ε
2860:δ
2809:∈
2757:ε
2751:δ
2728:∈
2705:ε
2662:ε
2612:⇒
2608:δ
2570:∈
2564:∀
2554:δ
2551:∃
2541:ε
2538:∀
2531:∈
2525:∀
2439:δ
2402:∈
2376:ε
2310:δ
2259:∈
2230:δ
2204:ε
2179:_
2085:∈
2079:∀
2056:∈
2050:∀
2024:δ
2021:∃
1995:ε
1992:∀
1968:ε
1918:⇒
1914:δ
1876:∈
1870:∀
1863:∈
1857:∀
1847:δ
1844:∃
1834:ε
1831:∀
1738:⟹
1714:ε
1691:−
1670:⟹
1666:δ
1652:−
1621:∈
1586:δ
1560:ε
1492:real line
1403:for each
1388:δ
1351:∈
1297:for each
1282:δ
1245:∈
1219:ε
1153:δ
1102:∈
1067:δ
1041:ε
902:→
701:ε
681:δ
661:ε
641:ϵ
621:δ
594:δ
534:ϵ
511:−
474:δ
454:ϵ
434:δ
414:δ
379:, a real
263:δ
223:ε
78:∈
75:δ
37:∈
34:ε
9180:Calculus
9118:(1976).
9050:(1960).
9023:(1989).
8972:See also
8873:we have
8548:implies
7004:we have
5403:Examples
3247:interval
2171:at
864:isometry
381:function
235:over an
8674:uniform
6762:. (see
6332:History
5638:but is
4886:and as
1980:. Here
1778:, then
1726:(where
1494:, then
546:at any
9134:
9086:
9067:
9035:
6326:graph.
5759:, and
5518:Every
5148:. The
4795:. But
4551:where
4120:, and
3251:global
2071:, and
9027:[
8963:Each
6338:Heine
4665:with
3346:below
3295:local
3183:local
2887:then
2274:with
1754:is a
1117:with
917:with
9132:ISBN
9084:ISBN
9065:ISBN
9033:ISBN
8617:For
8370:and
7789:>
7105:Let
6899:and
6264:>
6238:>
6154:>
6128:>
5977:and
5908:and
5864:>
5838:>
5529:The
5207:and
5128:The
5101:Any
5003:<
4977:<
4780:<
4706:<
4294:<
4240:<
4179:>
4166:>
4134:>
4105:>
4030:for
3994:for
3940:and
3904:are
3844:and
3783:<
3729:<
3681:>
3668:>
3508:and
3466:for
3161:and
2938:<
2857:<
2717:and
2659:<
2605:<
2557:>
2544:>
2436:<
2373:<
2307:<
2233:>
2207:>
2027:>
1998:>
1965:<
1911:<
1850:>
1837:>
1711:<
1663:<
1589:>
1563:>
1521:and
1470:and
1385:<
1279:<
1216:<
1150:<
1070:>
1044:>
959:and
713:and
566:and
531:<
88:>
47:>
9155:doi
8937:in
8853:in
8785:in
8741:in
8327:is
8092:to
7353:in
7265:If
7016:lim
6938:lim
6684:in
6540:in
6358:In
5792:as
5640:not
5216:lim
4573:is
4055:).
2798:if
1798:").
1450:If
1443:by
784:on
737:.
606:.
375:In
326:not
324:is
9171::
9151:32
9149:,
9130:.
9107:,
9101:,
8957:.
8660:.
8331:.
8226:.
7085:0.
5394:.
5125:.
4357:,
4091:,
3348:.
2973:,
2743:,
2042:,
2013:,
1636:,
1447:.
877:.
369:is
9157::
9140:.
9092:.
9073:.
9041:.
9006:.
8945:V
8925:)
8922:)
8917:2
8913:x
8909:(
8906:f
8903:,
8900:)
8895:1
8891:x
8887:(
8884:f
8881:(
8861:U
8841:)
8836:2
8832:x
8828:,
8823:1
8819:x
8815:(
8794:X
8772:U
8750:Y
8728:V
8700:Y
8694:X
8691::
8688:f
8640:W
8634:V
8631::
8628:f
8603:.
8600:B
8594:)
8589:2
8585:v
8581:(
8578:f
8572:)
8567:1
8563:v
8559:(
8556:f
8536:A
8528:2
8524:v
8515:1
8511:v
8490:V
8470:A
8450:W
8430:B
8410:W
8404:V
8401::
8398:f
8378:W
8358:V
8314:X
8292:X
8270:S
8249:R
8243:S
8240::
8237:f
8213:R
8191:f
8169:I
8147:I
8126:f
8106:I
8100:Q
8079:f
8057:I
8035:Q
8013:f
7988:)
7984:1
7972:a
7967:(
7961:x
7957:a
7953:=
7950:)
7947:x
7944:(
7941:f
7935:)
7929:+
7926:x
7923:(
7920:f
7897:R
7877:f
7852:x
7829:x
7825:a
7818:x
7815::
7812:f
7792:1
7786:a
7762:R
7739:2
7735:x
7728:x
7725:,
7722:R
7716:R
7713::
7710:f
7689:X
7664:f
7643:R
7637:X
7634::
7631:f
7610:f
7588:X
7566:Y
7544:X
7522:S
7501:X
7480:f
7455:R
7449:X
7446::
7443:f
7423:f
7403:X
7382:S
7361:X
7341:S
7320:f
7295:X
7274:S
7261:?
7249:X
7229:f
7207:R
7201:S
7198::
7195:f
7175:R
7166:,
7153:X
7133:S
7113:X
7082:=
7078:|
7074:)
7069:n
7065:y
7061:(
7058:f
7052:)
7047:n
7043:x
7039:(
7036:f
7032:|
7020:n
6989:0
6986:=
6982:|
6976:n
6972:y
6963:n
6959:x
6954:|
6942:n
6912:n
6908:y
6885:n
6881:x
6858:n
6853:R
6845:A
6842::
6839:f
6817:n
6812:R
6790:A
6778:(
6750:f
6724:f
6702:R
6672:A
6643:A
6623:f
6603:A
6597:a
6571:f
6549:R
6527:A
6505:f
6462:)
6459:a
6456:(
6447:f
6440:)
6434:+
6431:a
6428:(
6419:f
6397:a
6371:f
6310:2
6287:2
6267:0
6241:0
6200:2
6177:2
6157:0
6131:0
6089:2
6066:2
6046:)
6043:)
6040:x
6037:(
6034:f
6031:,
6028:x
6025:(
5985:y
5965:x
5925:)
5922:y
5919:(
5916:f
5896:)
5893:x
5890:(
5887:f
5867:0
5841:0
5818:.
5800:x
5772:x
5768:e
5745:x
5741:e
5718:x
5714:e
5707:x
5687:x
5676:.
5664:2
5660:/
5650:x
5626:)
5623:2
5619:/
5612:,
5609:2
5605:/
5595:(
5547:0
5544:=
5541:x
5500:]
5497:1
5494:,
5491:0
5488:[
5468:]
5465:1
5462:,
5459:0
5456:[
5433:b
5430:+
5427:x
5424:a
5418:x
5382:)
5378:R
5374:(
5369:0
5365:C
5358:)
5354:R
5350:(
5345:c
5341:C
5319:R
5298:)
5294:R
5290:(
5285:0
5281:C
5260:f
5240:)
5237:x
5234:(
5231:f
5220:x
5195:)
5189:,
5186:0
5183:[
5163:f
5086:f
5066:f
4973:|
4969:)
4966:x
4963:(
4960:f
4954:)
4948:+
4945:x
4942:(
4939:f
4935:|
4894:x
4871:,
4866:2
4858:+
4849:x
4846:2
4843:=
4840:)
4837:x
4834:(
4831:f
4824:)
4817:+
4814:x
4810:(
4806:f
4776:|
4772:)
4767:2
4763:x
4759:(
4756:f
4750:)
4745:1
4741:x
4737:(
4734:f
4730:|
4702:|
4696:2
4692:x
4683:1
4679:x
4674:|
4652:R
4643:2
4639:x
4635:,
4630:1
4626:x
4560:R
4537:2
4533:x
4526:x
4523:,
4519:R
4511:R
4504:f
4472:.
4466:x
4430:x
4387:,
4374:x
4335:,
4321:x
4309:.
4291:)
4288:)
4285:y
4282:(
4279:f
4276:,
4273:)
4270:x
4267:(
4264:f
4261:(
4256:2
4252:d
4237:)
4234:y
4231:,
4228:x
4225:(
4220:1
4216:d
4211::
4208:I
4202:y
4195:I
4189:x
4182:0
4169:0
4137:0
4108:0
4079:I
4073:x
4062:(
4042:R
4017:R
4009:R
4005::
4002:f
3981:|
3977:)
3974:y
3971:(
3968:f
3962:)
3959:x
3956:(
3953:f
3949:|
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836:=
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390:f
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160:=
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125:=
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119:x
116:(
113:f
91:0
83:R
72:2
50:0
42:R
31:2
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