2023:
736:
5508:
4027:
3699:
4974:
6043:
5287:
3807:
6700:
6499:. This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain. Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:
5838:
6361:
5503:{\displaystyle \Lambda (\beta )={\begin{bmatrix}{\frac {1}{\sqrt {1-\beta ^{2}}}}&-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&0&0\\-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}}
4022:{\displaystyle {\begin{aligned}r_{1}:\mathbb {R} \rightarrow \mathbb {R} &\quad r_{2}:\mathbb {R} \rightarrow \mathbb {R} &\cdots &\quad r_{n}:\mathbb {R} \rightarrow \mathbb {R} \\r_{1}=r_{1}(t)&\quad r_{2}=r_{2}(t)&\cdots &\quad r_{n}=r_{n}(t)\\\end{aligned}}}
6504:
4587:
4840:
3335:
3169:
5268:
6212:
4105:
6038:{\displaystyle \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=\mathrm {card} (\mathbb {R} )^{\mathrm {card} (\mathbb {R} )}={\mathfrak {c}}^{\mathfrak {c}}=(2^{\aleph _{0}})^{\mathfrak {c}}=2^{\aleph _{0}\cdot {\mathfrak {c}}}=2^{\mathfrak {c}}.}
6473:
4926:
5785:
2834:
2042:
between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.
5629:
5557:. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of a
6900:
6968:
6794:
4344:
4615:
6121:
1088:
Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subset
3211:
2995:
6695:{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\leq \mathrm {card} (\mathbb {R} ^{\mathbb {Q} })=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\cdot \aleph _{0}}=2^{\aleph _{0}}={\mathfrak {c}}.}
6176:
3029:
3812:
1281:
2034:
were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a
5178:
3508:
5827:
2233:
1842:
2664:
2533:
1778:
1016:
1963:
2468:
6832:
3664:
2297:
1651:
1226:
1157:
2587:
6205:
4038:
4224:
1321:
6497:
6356:{\displaystyle 2^{\mathfrak {c}}=\mathrm {card} (2^{\mathbb {R} })\leq \mathrm {card} (X^{\mathbb {R} })\leq \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=2^{\mathfrak {c}}.}
873:
792:
6724:
3693:
3555:
3458:
3432:
2127:
2070:
2008:
1894:
1710:
1587:
1466:
1186:
1112:
844:
814:
767:
703:
681:
659:
625:
587:
542:
516:
6371:
3596:
2372:
1379:
6065:
2333:
2253:
2157:
5716:
are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.
4855:
5731:
2711:
1041:
is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).
7102:
5575:
6837:
5646:
may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing
7135:
4582:{\displaystyle {\frac {r_{1}(t)-a_{1}}{dr_{1}(t)/dt}}={\frac {r_{2}(t)-a_{2}}{dr_{2}(t)/dt}}=\cdots ={\frac {r_{n}(t)-a_{n}}{dr_{n}(t)/dt}}}
6905:
7128:
6729:
4835:{\displaystyle (p_{1}-a_{1}){\frac {dr_{1}(t)}{dt}}+(p_{2}-a_{2}){\frac {dr_{2}(t)}{dt}}+\cdots +(p_{n}-a_{n}){\frac {dr_{n}(t)}{dt}}=0}
960:
Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:
895:
at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:
7165:
6070:
3330:{\displaystyle \int _{a}^{b}\mathbf {y} (x)\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {y} (x)\cdot {\frac {d\mathbf {r} (x)}{dx}}dx}
3381:
567:
Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of
3164:{\displaystyle {\frac {d\mathbf {y} }{dx}}=\left({\frac {dy_{1}}{dx}},{\frac {dy_{2}}{dx}},\ldots ,{\frac {dy_{n}}{dx}}\right)}
2849:
6126:
7091:
7072:
6994:
238:
1232:
448:
5263:{\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}
6984:
193:
7112:
7023:
5153:
3466:
1397:. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an
7217:
7048:
5790:
7396:
7297:
7224:
3369:
2178:
1871:
1033:
Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:
724:
58:
5634:
where one takes the derivative of a wave function, which can be an element of several different
Hilbert spaces.
1783:
7386:
7207:
424:
223:
2606:
2487:
1715:
982:
7212:
7158:
3560:
is an equation in the variables. Implicit functions are a more general way to represent functions, since if:
17:
1902:
2419:
4100:{\displaystyle \mathbf {r} :\mathbb {R} \rightarrow \mathbb {R} ^{n}\,,\quad \mathbf {r} =\mathbf {r} (t)}
6799:
4291:, the equations of the one-dimensional tangent line to the curve at that point are given in terms of the
3669:
but the converse is not always possible, i.e. not all implicit functions have the form of this equation.
3607:
2258:
1867:
1603:
1530:
Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by
1197:
1128:
2548:
887:
For many commonly used real functions, the domain is the whole set of real numbers, and the function is
6181:
908:
258:
4120:
1287:
6478:
1538:. This means that it is not worthy to explicitly define the domain of a function of a real variable.
1081:). For simplicity, in this article a real-valued function of a real variable will be simply called a
941:
Some functions are defined and continuous everywhere, but not everywhere differentiable. For example
346:
1085:. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.
7151:
3368:
With the definitions of integration and derivatives, key theorems can be formulated, including the
853:
772:
6707:
3676:
3672:
3519:
3441:
3415:
2075:
2053:
1991:
1877:
1693:
1560:
1449:
1169:
1095:
827:
797:
750:
727:, the function may be viewed as a sequence of real functions. This is often used in applications.
686:
664:
642:
608:
570:
525:
499:
5549:
Generalizing the previous section, the output of a function of a real variable can also lie in a
934:
892:
549:
441:
208:
125:
80:
5566:
2596:
is in the interior of the domain, the limit exists if and only if the function is continuous at
7329:
7229:
6468:{\displaystyle C^{0}(\mathbb {R} )=\{f:\mathbb {R} \to \mathbb {R} :f\ \mathrm {continuous} \}}
5830:
5278:
1398:
1119:
1063:
1055:
876:
744:
545:
486:
33:
2022:
7391:
7322:
7317:
7281:
7277:
7202:
7175:
1535:
462:
404:
3566:
1122:
of positive length. In other words, a real-valued function of a real variable is a function
7349:
7254:
5558:
5165:
3373:
2345:
1443:
1115:
923:
821:
628:
602:
557:
490:
389:
351:
155:
66:
1355:
8:
7344:
7334:
7287:
6366:
4292:
3377:
2674:
2378:
2160:
2031:
2011:
1531:
1349:
1343:
900:
888:
717:
713:
709:
470:
323:
313:
308:
7143:
956:
is defined and continuous everywhere, and is differentiable everywhere, except for zero.
949:
is defined and continuous everywhere, and is differentiable everywhere, except for zero.
7366:
7249:
6050:
5833:(i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic:
5562:
5274:
2386:
2318:
2238:
2142:
1026:
969:
434:
318:
293:
929:
Some functions are defined everywhere, but not continuous at some points. For example
7339:
7302:
7108:
7087:
7068:
7044:
7019:
5530:
3395:
2047:
2035:
1555:
1402:
965:
904:
598:
394:
298:
288:
273:
268:
3721:
the red line is the tangent to the curve, and the blue plane is normal to the curve.
7292:
7272:
6989:
976:
384:
361:
110:
7307:
7244:
7239:
7234:
5273:
is a matrix valued function of rotation angle of about the origin. Similarly, in
5169:
5033:
4921:{\displaystyle (\mathbf {p} -\mathbf {a} )\cdot {\frac {d\mathbf {r} (t)}{dt}}=0}
3703:
3380:. Evaluating a mixture of integrals and derivatives can be done by using theorem
3187:
594:
478:
379:
356:
303:
1546:
The arithmetic operations may be applied to the functions in the following way:
735:
5647:
5538:
2039:
946:
632:
399:
5780:{\displaystyle \mathbb {R} ^{\mathbb {R} }=\{f:\mathbb {R} \to \mathbb {R} \}}
2829:{\displaystyle y_{1}=f_{1}(x)\,,\quad y_{2}=f_{2}(x)\,,\ldots ,y_{n}=f_{n}(x)}
7380:
7311:
7267:
6979:
5554:
3175:
716:
in the codomain. In this context, a function that defines curve is called a
548:
of positive length. Most real functions that are considered and studied are
5624:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }
5550:
5149:
3435:
2374:
A function is continuous if it is continuous at every point of its domain.
1468:
that is sometimes explicitly defined. In fact, if one restricts the domain
590:
95:
5152:" of the particle, and is a vector normal to the curve directed along the
5725:
4846:
4230:
3341:
3179:
1327:
1059:
1038:
817:
494:
474:
6895:{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\geq {\mathfrak {c}}}
2473:
if the following condition is satisfied: For every positive real number
7197:
953:
636:
283:
278:
6475:
has a strictly smaller cardinality, the cardinality of the continuum,
5719:
968:
is a quotient of two polynomial functions, and is not defined at the
328:
6963:{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))={\mathfrak {c}}}
2689:, the latter formula allows to "extend by continuity" the domain of
560:
of a real variable, that is, the functions of a real variable whose
552:
in some interval. The most widely considered such functions are the
7354:
7192:
7187:
6789:{\displaystyle \{f:\mathbb {R} \to \mathbb {R} :f(x)\equiv x_{0}\}}
5017:
5013:
2705:
One can collect a number of functions each of a real variable, say
1417:
1409:
847:
683:-vector space on the functions. If the codomain has a structure of
561:
466:
409:
140:
70:
5544:
414:
2381:
of a real-valued function of a real variable is as follows. Let
2072:, which is an everywhere defined function of 2 real variables:
917:
519:
5168:
can also be a function of a single variable. For example, the
3698:
2010:. This constraint implies that the above two algebras are not
5637:
5533:
between the frames of reference (a continuous variable), and
4973:
6704:
On the other hand, since there is a clear bijection between
6116:{\displaystyle 2\leq \mathrm {card} (X)\leq {\mathfrak {c}}}
5664:
is such a complex valued function, it may be decomposed as
195:
5565:. This occurs, for instance, in the general time-dependent
1862:
variables that are everywhere defined and the functions of
913:
2046:
For defining the continuity, it is useful to consider the
7173:
2990:{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})=}
1401:
or a single value. In the latter case, the function is a
1191:
A simple example of a function in one variable could be:
6171:{\displaystyle X^{\mathbb {R} }=\{f:\mathbb {R} \to X\}}
5728:
of the set of real-valued functions of a real variable,
1118:
of the function, which is always supposed to contain an
723:
When the codomain of a function of a real variable is a
7086:. Schaum's outline series (3rd ed.). McGraw Hill.
7081:
7067:. Schaum's outline series (5th ed.). McGraw Hill.
7018:. Vol. 2. Wiley Classics Library. pp. 46β47.
5829:, which is strictly larger than the cardinality of the
1446:
of a function of several real variables is a subset of
7062:
5311:
5202:
4601:-dimensional hyperplane normal to the tangent line at
6908:
6840:
6802:
6732:
6710:
6507:
6481:
6374:
6215:
6184:
6129:
6073:
6053:
5841:
5793:
5734:
5578:
5290:
5181:
5040:, and is a vector tangent to the space curve for all
4858:
4618:
4347:
4123:
4041:
3810:
3679:
3610:
3569:
3522:
3469:
3444:
3418:
3214:
3032:
2852:
2714:
2609:
2551:
2490:
2422:
2348:
2321:
2261:
2241:
2181:
2145:
2078:
2056:
1994:
1905:
1880:
1786:
1718:
1696:
1606:
1563:
1452:
1358:
1290:
1235:
1200:
1172:
1131:
1098:
985:
856:
830:
800:
775:
753:
689:
667:
661:-vector space of the codomain induces a structure of
645:
611:
573:
528:
502:
1029:
is defined only for positive values of the variable.
875:
It is generally assumed that the domain contains an
5720:
Cardinality of sets of functions of a real variable
4977:Kinematic quantities of a classical particle: mass
2305:may be chosen small enough for having the image by
1506:. In practice, it is often not harmful to identify
1276:{\displaystyle X=\{x\in \mathbb {R} \,:\,x\geq 0\}}
116:
6962:
6894:
6826:
6788:
6718:
6694:
6491:
6467:
6355:
6199:
6170:
6115:
6059:
6037:
5821:
5779:
5623:
5502:
5262:
5079:, and the hyperplane normal to the space curve at
4920:
4834:
4581:
4218:
4099:
4021:
3687:
3658:
3590:
3549:
3502:
3452:
3426:
3329:
3163:
2989:
2828:
2658:
2581:
2527:
2462:
2366:
2327:
2291:
2247:
2227:
2163:to its domain, if, for every positive real number
2151:
2121:
2064:
2002:
1965:which is a function only if the set of the points
1957:
1888:
1836:
1772:
1704:
1645:
1581:
1460:
1373:
1315:
1275:
1220:
1180:
1151:
1106:
1010:
937:is defined everywhere, but not continuous at zero.
867:
838:
808:
786:
761:
697:
675:
653:
619:
581:
536:
510:
112:
7378:
2626:
2430:
240:
7104:Functions of a Real Variable: Elementary Theory
5545:Banach and Hilbert spaces and quantum mechanics
3503:{\displaystyle \phi :\mathbb {R} ^{2}\to \{0\}}
3201:), by integrating with respect to the variable
593:over the reals. That is, the codomain may be a
1484:, one gets formally a different function, the
1188:that contains an interval of positive length.
705:-algebra, the same is true for the functions.
7159:
7138:Differentiability for Multivariable Functions
5281:matrix for a pure boost (without rotations):
5044:in the instantaneous direction of motion. At
4997:The physical and geometric interpretation of
442:
6783:
6733:
6462:
6399:
6165:
6145:
5822:{\displaystyle \beth _{2}=2^{\mathfrak {c}}}
5774:
5752:
3497:
3491:
2030:Until the second part of 19th century, only
2026:Limit of a real function of a real variable.
1844:are functions that have a domain containing
1270:
1242:
88:
7100:
3412:)". Instead, the mapping is from the space
2228:{\displaystyle |f(x)-f(a)|<\varepsilon }
7166:
7152:
5644:complex-valued function of a real variable
5638:Complex-valued function of a real variable
5159:
3360:are the start and endpoints of the curve.
1837:{\displaystyle f\,g:(x)\mapsto f(x)\,g(x)}
449:
435:
97:
7043:. New York: McGraw-Hill. pp. 98β99.
6940:
6872:
6817:
6751:
6743:
6712:
6577:
6571:
6539:
6417:
6409:
6389:
6326:
6320:
6289:
6254:
6155:
6136:
5924:
5897:
5867:
5861:
5770:
5762:
5743:
5737:
4968:
4070:
4060:
4051:
3900:
3892:
3866:
3858:
3837:
3829:
3681:
3478:
3446:
3420:
2784:
2747:
2058:
1996:
1882:
1821:
1790:
1698:
1454:
1260:
1256:
1252:
1214:
1174:
1145:
1100:
1069:, for producing another real number, the
858:
832:
802:
777:
755:
691:
669:
647:
613:
575:
530:
504:
4972:
4592:
4236:
3697:
2659:{\displaystyle f(a)=\lim _{x\to a}f(x).}
2528:{\displaystyle |f(x)-L|<\varepsilon }
2477:> 0, there is a positive real number
2021:
1773:{\displaystyle f+g:(x)\mapsto f(x)+g(x)}
1672:are two functions of respective domains
1011:{\displaystyle {\frac {\pi }{2}}+k\pi ,}
734:
5052:, the space curve has a tangent vector
3382:differentiation under the integral sign
2017:
1052:real-valued function of a real variable
605:of real numbers of a given size, or an
225:
216:
14:
7379:
7013:
2592:If the limit exists, it is unique. If
1958:{\displaystyle 1/f:(x)\mapsto 1/f(x),}
1541:
712:of a function of a real variable is a
7147:
7038:
6995:Function of several complex variables
5513:is a function of the boost parameter
3387:
2463:{\displaystyle L=\lim _{x\to a}f(x),}
1045:
244:
229:
210:
199:
183:
157:
146:
127:
101:
82:
2335:contained in the interval of length
179:
163:
7041:Principles of Mathematical Analysis
6955:
6887:
6827:{\displaystyle C^{0}(\mathbb {R} )}
6684:
6484:
6344:
6222:
6191:
6108:
6026:
6010:
5981:
5946:
5939:
5813:
3659:{\displaystyle \phi (x,y)=y-f(x)=0}
2292:{\displaystyle d(x,a)<\varphi .}
1866:variables that are defined in some
1646:{\displaystyle rf:(x)\mapsto rf(x)}
1416:is the set of the solutions of the
1221:{\displaystyle f:X\to \mathbb {R} }
1152:{\displaystyle f:X\to \mathbb {R} }
24:
7016:Differential and Integral Calculus
6985:Function of several real variables
6919:
6916:
6913:
6910:
6851:
6848:
6845:
6842:
6726:and the set of constant functions
6668:
6648:
6635:
6615:
6598:
6562:
6559:
6556:
6553:
6518:
6515:
6512:
6509:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6434:
6431:
6311:
6308:
6305:
6302:
6276:
6273:
6270:
6267:
6241:
6238:
6235:
6232:
6123:, then the cardinality of the set
6090:
6087:
6084:
6081:
5996:
5964:
5916:
5913:
5910:
5907:
5889:
5886:
5883:
5880:
5852:
5849:
5846:
5843:
5618:
5600:
5591:
5587:
5291:
2582:{\displaystyle d(x,a)<\delta .}
2169:, there is a positive real number
1073:of the function, commonly denoted
25:
7408:
7122:
6200:{\displaystyle 2^{\mathfrak {c}}}
5582:
5087:is also normal to the tangent at
5036:coordinates parametrized by time
1858:It follows that the functions of
882:
7082:R. Wrede, M. R. Spiegel (2010).
4888:
4871:
4863:
4219:{\displaystyle \mathbf {r} (t)=}
4125:
4084:
4076:
4043:
3673:One-dimensional space curves in
3297:
3274:
3251:
3231:
3040:
2854:
1316:{\displaystyle f(x)={\sqrt {x}}}
730:
142:
133:
7298:Least-squares spectral analysis
7225:Fundamental theorem of calculus
7063:F. Ayres, E. Mendelson (2009).
6492:{\displaystyle {\mathfrak {c}}}
4074:
3982:
3942:
3877:
3843:
3370:fundamental theorem of calculus
2751:
824:of a real function is a subset
725:finite-dimensional vector space
59:History of the function concept
7032:
7007:
6947:
6944:
6936:
6923:
6879:
6876:
6868:
6855:
6821:
6813:
6767:
6761:
6747:
6610:
6589:
6583:
6566:
6546:
6543:
6535:
6522:
6413:
6393:
6385:
6332:
6315:
6295:
6280:
6260:
6245:
6159:
6100:
6094:
5976:
5955:
5928:
5920:
5902:
5893:
5873:
5856:
5766:
5612:
5300:
5294:
5191:
5185:
4898:
4892:
4875:
4859:
4812:
4806:
4787:
4761:
4738:
4732:
4713:
4687:
4670:
4664:
4645:
4619:
4562:
4556:
4525:
4519:
4480:
4474:
4443:
4437:
4404:
4398:
4367:
4361:
4213:
4210:
4204:
4182:
4176:
4160:
4154:
4141:
4135:
4129:
4094:
4088:
4055:
4012:
4006:
3972:
3966:
3937:
3931:
3896:
3862:
3833:
3725:
3647:
3641:
3626:
3614:
3585:
3579:
3538:
3526:
3488:
3307:
3301:
3284:
3278:
3241:
3235:
2984:
2981:
2975:
2953:
2947:
2931:
2925:
2912:
2906:
2861:
2839:into a vector parametrized by
2823:
2817:
2781:
2775:
2744:
2738:
2650:
2644:
2633:
2619:
2613:
2567:
2555:
2515:
2505:
2499:
2492:
2454:
2448:
2437:
2358:
2352:
2277:
2265:
2215:
2211:
2205:
2196:
2190:
2183:
2115:
2101:
2094:
2082:
1949:
1943:
1929:
1926:
1920:
1831:
1825:
1818:
1812:
1806:
1803:
1797:
1767:
1761:
1752:
1746:
1740:
1737:
1731:
1640:
1634:
1625:
1622:
1616:
1573:
1570:
1564:
1368:
1362:
1300:
1294:
1210:
1141:
1062:, commonly represented by the
13:
1:
7000:
3004:is the vector derivatives of
3000:The derivative of the vector
1657:(or is everywhere defined if
868:{\displaystyle \mathbb {R} .}
787:{\displaystyle \mathbb {R} ,}
6719:{\displaystyle \mathbb {R} }
5095:. Any vector in this plane (
4229:describes a one-dimensional
3709:is parametrized by a scalar
3688:{\displaystyle \mathbb {R} }
3550:{\displaystyle \phi (x,y)=0}
3460:(just the ordinary zero 0):
3453:{\displaystyle \mathbb {R} }
3427:{\displaystyle \mathbb {R} }
3400:is not written in the form "
2122:{\displaystyle d(x,y)=|x-y|}
2065:{\displaystyle \mathbb {R} }
2003:{\displaystyle \mathbb {R} }
1889:{\displaystyle \mathbb {R} }
1705:{\displaystyle \mathbb {R} }
1582:{\displaystyle (x)\mapsto r}
1520:, and to omit the subscript
1461:{\displaystyle \mathbb {R} }
1391:runs in the whole domain of
1381:is the set of all values of
1181:{\displaystyle \mathbb {R} }
1107:{\displaystyle \mathbb {R} }
839:{\displaystyle \mathbb {R} }
816:denotes as usual the set of
809:{\displaystyle \mathbb {R} }
762:{\displaystyle \mathbb {R} }
739:The graph of a real function
698:{\displaystyle \mathbb {R} }
676:{\displaystyle \mathbb {R} }
654:{\displaystyle \mathbb {R} }
620:{\displaystyle \mathbb {R} }
582:{\displaystyle \mathbb {R} }
564:is the set of real numbers.
537:{\displaystyle \mathbb {R} }
511:{\displaystyle \mathbb {R} }
7:
6973:
3601:then we can always define:
3363:
2700:
1988:contains an open subset of
1870:of a given point both form
1690:contains an open subset of
483:function of a real variable
10:
7413:
6796:, which forms a subset of
4965:, not on the space curve.
2309:of the interval of radius
1341:
425:List of specific functions
7363:
7263:
7182:
3797:all of a common variable
2600:. In this case, we have
1899:One may similarly define
1437:
6902:must also hold. Hence,
2542:in the domain such that
1589:, is everywhere defined.
1337:
5160:Matrix valued functions
3702:Space curve in 3d. The
1653:has the same domain as
1412:of a given real number
935:Heaviside step function
7397:Multivariable calculus
7230:Calculus of variations
7203:Differential equations
7130:Multivariable Calculus
6964:
6896:
6828:
6790:
6720:
6696:
6493:
6469:
6357:
6201:
6172:
6117:
6061:
6039:
5823:
5781:
5625:
5504:
5279:Lorentz transformation
5264:
5020:moving along the path
4994:
4969:Relation to kinematics
4922:
4836:
4583:
4220:
4110:then the parametrized
4101:
4023:
3722:
3689:
3660:
3592:
3591:{\displaystyle y=f(x)}
3551:
3504:
3454:
3428:
3331:
3165:
2991:
2830:
2660:
2583:
2529:
2464:
2368:
2329:
2293:
2249:
2229:
2153:
2123:
2066:
2027:
2004:
1959:
1890:
1838:
1774:
1706:
1647:
1592:For every real number
1583:
1550:For every real number
1462:
1375:
1317:
1277:
1222:
1182:
1153:
1108:
1058:that takes as input a
1012:
869:
840:
810:
788:
763:
740:
699:
677:
655:
621:
583:
538:
512:
465:, and applications in
7387:Mathematical analysis
7323:Representation theory
7282:quaternionic analysis
7278:Hypercomplex analysis
7176:mathematical analysis
6965:
6897:
6829:
6791:
6721:
6697:
6494:
6470:
6358:
6202:
6173:
6118:
6062:
6040:
5824:
5782:
5626:
5505:
5265:
4976:
4923:
4837:
4593:Normal plane to curve
4584:
4237:Tangent line to curve
4221:
4102:
4024:
3701:
3690:
3661:
3593:
3552:
3505:
3455:
3429:
3332:
3174:One can also perform
3166:
2992:
2831:
2661:
2584:
2530:
2465:
2369:
2367:{\displaystyle f(a).}
2330:
2294:
2250:
2230:
2154:
2124:
2067:
2025:
2005:
1960:
1891:
1839:
1775:
1707:
1648:
1584:
1536:analytic continuation
1463:
1376:
1318:
1278:
1223:
1183:
1162:such that its domain
1154:
1109:
1013:
870:
841:
811:
789:
764:
743:A real function is a
738:
700:
678:
656:
622:
584:
558:real-valued functions
539:
513:
463:mathematical analysis
27:Mathematical function
7255:Table of derivatives
7136:L. A. Talman (2007)
7101:N. Bourbaki (2004).
6906:
6838:
6800:
6730:
6708:
6505:
6479:
6372:
6367:continuous functions
6365:However, the set of
6213:
6182:
6127:
6071:
6051:
5839:
5791:
5732:
5576:
5567:SchrΓΆdinger equation
5288:
5179:
5103:) must be normal to
4856:
4616:
4597:The equation of the
4345:
4293:ordinary derivatives
4121:
4039:
3808:
3730:Given the functions
3677:
3608:
3567:
3520:
3467:
3442:
3416:
3374:integration by parts
3212:
3030:
2850:
2712:
2607:
2549:
2488:
2420:
2346:
2319:
2259:
2239:
2179:
2143:
2076:
2054:
2032:continuous functions
2018:Continuity and limit
1992:
1903:
1878:
1872:commutative algebras
1784:
1716:
1694:
1604:
1561:
1450:
1374:{\displaystyle f(x)}
1356:
1288:
1233:
1198:
1170:
1129:
1096:
983:
924:Exponential function
901:polynomial functions
879:of positive length.
854:
828:
798:
773:
751:
687:
665:
643:
609:
571:
526:
500:
7335:Continuous function
7288:Functional analysis
6067:is a set such that
5154:radius of curvature
4845:or in terms of the
4032:or taken together:
3272:
3229:
2387:topological closure
1596:and every function
1542:Algebraic structure
1496:, which is denoted
1344:Image (mathematics)
979:is not defined for
972:of the denominator.
718:parametric equation
471:applied mathematics
7367:Mathematics portal
7250:Lists of integrals
7039:Rudin, W. (1976).
6960:
6892:
6824:
6786:
6716:
6692:
6489:
6465:
6353:
6197:
6168:
6113:
6057:
6035:
5819:
5777:
5621:
5500:
5494:
5275:special relativity
5260:
5254:
5016:" of a point-like
4995:
4918:
4832:
4579:
4283:for some constant
4216:
4097:
4019:
4017:
3723:
3685:
3656:
3588:
3547:
3500:
3450:
3424:
3398:of a real variable
3388:Implicit functions
3327:
3258:
3215:
3161:
2987:
2826:
2656:
2640:
2579:
2525:
2481:> 0 such that
2460:
2444:
2364:
2325:
2289:
2245:
2225:
2149:
2119:
2062:
2028:
2000:
1955:
1886:
1834:
1770:
1702:
1643:
1579:
1458:
1387:when the variable
1371:
1313:
1273:
1218:
1178:
1149:
1104:
1046:General definition
1027:logarithm function
1008:
905:constant functions
865:
836:
806:
784:
759:
741:
695:
673:
651:
617:
579:
534:
508:
259:Classes/properties
7374:
7373:
7340:Special functions
7303:Harmonic analysis
7093:978-0-07-162366-7
7084:Advanced calculus
7074:978-0-07-150861-2
6429:
6060:{\displaystyle X}
5615:
5598:
5531:relative velocity
5436:
5435:
5409:
5408:
5367:
5366:
5337:
5336:
4910:
4824:
4750:
4682:
4577:
4495:
4419:
3396:implicit function
3319:
3154:
3118:
3088:
3053:
2677:of the domain of
2625:
2429:
2328:{\displaystyle a}
2248:{\displaystyle x}
2152:{\displaystyle a}
2048:distance function
2036:topological space
1973:in the domain of
1556:constant function
1403:constant function
1311:
994:
966:rational function
747:from a subset of
599:coordinate vector
544:that contains an
459:
458:
371:Generalizations
16:(Redirected from
7404:
7293:Fourier analysis
7273:Complex analysis
7174:Major topics in
7168:
7161:
7154:
7145:
7144:
7118:
7097:
7078:
7055:
7054:
7036:
7030:
7029:
7011:
6990:Complex analysis
6969:
6967:
6966:
6961:
6959:
6958:
6943:
6935:
6934:
6922:
6901:
6899:
6898:
6893:
6891:
6890:
6875:
6867:
6866:
6854:
6833:
6831:
6830:
6825:
6820:
6812:
6811:
6795:
6793:
6792:
6787:
6782:
6781:
6754:
6746:
6725:
6723:
6722:
6717:
6715:
6701:
6699:
6698:
6693:
6688:
6687:
6678:
6677:
6676:
6675:
6658:
6657:
6656:
6655:
6643:
6642:
6625:
6624:
6623:
6622:
6608:
6607:
6606:
6605:
6582:
6581:
6580:
6574:
6565:
6542:
6534:
6533:
6521:
6498:
6496:
6495:
6490:
6488:
6487:
6474:
6472:
6471:
6466:
6461:
6427:
6420:
6412:
6392:
6384:
6383:
6362:
6360:
6359:
6354:
6349:
6348:
6347:
6331:
6330:
6329:
6323:
6314:
6294:
6293:
6292:
6279:
6259:
6258:
6257:
6244:
6227:
6226:
6225:
6206:
6204:
6203:
6198:
6196:
6195:
6194:
6177:
6175:
6174:
6169:
6158:
6141:
6140:
6139:
6122:
6120:
6119:
6114:
6112:
6111:
6093:
6066:
6064:
6063:
6058:
6047:Furthermore, if
6044:
6042:
6041:
6036:
6031:
6030:
6029:
6016:
6015:
6014:
6013:
6004:
6003:
5986:
5985:
5984:
5974:
5973:
5972:
5971:
5951:
5950:
5949:
5943:
5942:
5932:
5931:
5927:
5919:
5900:
5892:
5872:
5871:
5870:
5864:
5855:
5828:
5826:
5825:
5820:
5818:
5817:
5816:
5803:
5802:
5786:
5784:
5783:
5778:
5773:
5765:
5748:
5747:
5746:
5740:
5715:
5709:
5699:
5688:
5677:
5663:
5630:
5628:
5627:
5622:
5617:
5616:
5608:
5599:
5597:
5586:
5509:
5507:
5506:
5501:
5499:
5498:
5437:
5434:
5433:
5418:
5414:
5410:
5407:
5406:
5391:
5387:
5368:
5365:
5364:
5349:
5345:
5338:
5335:
5334:
5319:
5315:
5269:
5267:
5266:
5261:
5259:
5258:
5129:
5078:
4960:
4927:
4925:
4924:
4919:
4911:
4909:
4901:
4891:
4882:
4874:
4866:
4841:
4839:
4838:
4833:
4825:
4823:
4815:
4805:
4804:
4791:
4786:
4785:
4773:
4772:
4751:
4749:
4741:
4731:
4730:
4717:
4712:
4711:
4699:
4698:
4683:
4681:
4673:
4663:
4662:
4649:
4644:
4643:
4631:
4630:
4588:
4586:
4585:
4580:
4578:
4576:
4569:
4555:
4554:
4541:
4540:
4539:
4518:
4517:
4507:
4496:
4494:
4487:
4473:
4472:
4459:
4458:
4457:
4436:
4435:
4425:
4420:
4418:
4411:
4397:
4396:
4383:
4382:
4381:
4360:
4359:
4349:
4334:with respect to
4282:
4225:
4223:
4222:
4217:
4203:
4202:
4175:
4174:
4153:
4152:
4128:
4106:
4104:
4103:
4098:
4087:
4079:
4069:
4068:
4063:
4054:
4046:
4028:
4026:
4025:
4020:
4018:
4005:
4004:
3992:
3991:
3965:
3964:
3952:
3951:
3930:
3929:
3917:
3916:
3903:
3895:
3887:
3886:
3869:
3861:
3853:
3852:
3840:
3832:
3824:
3823:
3796:
3771:
3750:
3694:
3692:
3691:
3686:
3684:
3665:
3663:
3662:
3657:
3597:
3595:
3594:
3589:
3556:
3554:
3553:
3548:
3509:
3507:
3506:
3501:
3487:
3486:
3481:
3459:
3457:
3456:
3451:
3449:
3433:
3431:
3430:
3425:
3423:
3378:Taylor's theorem
3336:
3334:
3333:
3328:
3320:
3318:
3310:
3300:
3291:
3277:
3271:
3266:
3254:
3234:
3228:
3223:
3182:parametrized by
3170:
3168:
3167:
3162:
3160:
3156:
3155:
3153:
3145:
3144:
3143:
3130:
3119:
3117:
3109:
3108:
3107:
3094:
3089:
3087:
3079:
3078:
3077:
3064:
3054:
3052:
3044:
3043:
3034:
2996:
2994:
2993:
2988:
2974:
2973:
2946:
2945:
2924:
2923:
2905:
2904:
2886:
2885:
2873:
2872:
2857:
2835:
2833:
2832:
2827:
2816:
2815:
2803:
2802:
2774:
2773:
2761:
2760:
2737:
2736:
2724:
2723:
2665:
2663:
2662:
2657:
2639:
2588:
2586:
2585:
2580:
2534:
2532:
2531:
2526:
2518:
2495:
2469:
2467:
2466:
2461:
2443:
2397:. The function,
2393:of the function
2373:
2371:
2370:
2365:
2341:
2334:
2332:
2331:
2326:
2314:
2304:
2299:In other words,
2298:
2296:
2295:
2290:
2254:
2252:
2251:
2246:
2234:
2232:
2231:
2226:
2218:
2186:
2174:
2168:
2158:
2156:
2155:
2150:
2128:
2126:
2125:
2120:
2118:
2104:
2071:
2069:
2068:
2063:
2061:
2009:
2007:
2006:
2001:
1999:
1987:
1972:
1964:
1962:
1961:
1956:
1939:
1913:
1895:
1893:
1892:
1887:
1885:
1874:over the reals (
1853:
1843:
1841:
1840:
1835:
1779:
1777:
1776:
1771:
1711:
1709:
1708:
1703:
1701:
1689:
1652:
1650:
1649:
1644:
1588:
1586:
1585:
1580:
1467:
1465:
1464:
1459:
1457:
1433:
1396:
1386:
1380:
1378:
1377:
1372:
1322:
1320:
1319:
1314:
1312:
1307:
1282:
1280:
1279:
1274:
1255:
1227:
1225:
1224:
1219:
1217:
1187:
1185:
1184:
1179:
1177:
1158:
1156:
1155:
1150:
1148:
1113:
1111:
1110:
1105:
1103:
1021:
1017:
1015:
1014:
1009:
995:
987:
977:tangent function
909:linear functions
874:
872:
871:
866:
861:
845:
843:
842:
837:
835:
815:
813:
812:
807:
805:
793:
791:
790:
785:
780:
768:
766:
765:
760:
758:
704:
702:
701:
696:
694:
682:
680:
679:
674:
672:
660:
658:
657:
652:
650:
639:. The structure
626:
624:
623:
618:
616:
588:
586:
585:
580:
578:
556:, which are the
543:
541:
540:
535:
533:
517:
515:
514:
509:
507:
479:natural sciences
451:
444:
437:
249:
248:
242:
234:
233:
227:
219:
218:
214:
204:
203:
197:
188:
187:
181:
166:
165:
161:
151:
150:
144:
136:
135:
131:
121:
120:
114:
106:
105:
99:
91:
90:
86:
53:
30:
29:
21:
7412:
7411:
7407:
7406:
7405:
7403:
7402:
7401:
7377:
7376:
7375:
7370:
7359:
7308:P-adic analysis
7259:
7245:Matrix calculus
7240:Tensor calculus
7235:Vector calculus
7198:Differentiation
7178:
7172:
7125:
7115:
7094:
7075:
7059:
7058:
7051:
7037:
7033:
7026:
7012:
7008:
7003:
6976:
6954:
6953:
6939:
6930:
6926:
6909:
6907:
6904:
6903:
6886:
6885:
6871:
6862:
6858:
6841:
6839:
6836:
6835:
6816:
6807:
6803:
6801:
6798:
6797:
6777:
6773:
6750:
6742:
6731:
6728:
6727:
6711:
6709:
6706:
6705:
6683:
6682:
6671:
6667:
6666:
6662:
6651:
6647:
6638:
6634:
6633:
6629:
6618:
6614:
6613:
6609:
6601:
6597:
6596:
6592:
6576:
6575:
6570:
6569:
6552:
6538:
6529:
6525:
6508:
6506:
6503:
6502:
6483:
6482:
6480:
6477:
6476:
6430:
6416:
6408:
6388:
6379:
6375:
6373:
6370:
6369:
6343:
6342:
6338:
6325:
6324:
6319:
6318:
6301:
6288:
6287:
6283:
6266:
6253:
6252:
6248:
6231:
6221:
6220:
6216:
6214:
6211:
6210:
6190:
6189:
6185:
6183:
6180:
6179:
6154:
6135:
6134:
6130:
6128:
6125:
6124:
6107:
6106:
6080:
6072:
6069:
6068:
6052:
6049:
6048:
6025:
6024:
6020:
6009:
6008:
5999:
5995:
5994:
5990:
5980:
5979:
5975:
5967:
5963:
5962:
5958:
5945:
5944:
5938:
5937:
5936:
5923:
5906:
5905:
5901:
5896:
5879:
5866:
5865:
5860:
5859:
5842:
5840:
5837:
5836:
5812:
5811:
5807:
5798:
5794:
5792:
5789:
5788:
5769:
5761:
5742:
5741:
5736:
5735:
5733:
5730:
5729:
5722:
5711:
5705:
5690:
5679:
5668:
5654:
5640:
5607:
5606:
5590:
5585:
5577:
5574:
5573:
5547:
5493:
5492:
5487:
5482:
5477:
5471:
5470:
5465:
5460:
5455:
5449:
5448:
5443:
5438:
5429:
5425:
5413:
5411:
5402:
5398:
5386:
5380:
5379:
5374:
5369:
5360:
5356:
5344:
5339:
5330:
5326:
5314:
5307:
5306:
5289:
5286:
5285:
5253:
5252:
5241:
5229:
5228:
5214:
5198:
5197:
5180:
5177:
5176:
5170:rotation matrix
5162:
5128:
5104:
5077:
5053:
5034:position vector
5032:as the spatial
4989:, acceleration
4971:
4958:
4949:
4942:
4932:
4902:
4887:
4883:
4881:
4870:
4862:
4857:
4854:
4853:
4816:
4800:
4796:
4792:
4790:
4781:
4777:
4768:
4764:
4742:
4726:
4722:
4718:
4716:
4707:
4703:
4694:
4690:
4674:
4658:
4654:
4650:
4648:
4639:
4635:
4626:
4622:
4617:
4614:
4613:
4595:
4565:
4550:
4546:
4542:
4535:
4531:
4513:
4509:
4508:
4506:
4483:
4468:
4464:
4460:
4453:
4449:
4431:
4427:
4426:
4424:
4407:
4392:
4388:
4384:
4377:
4373:
4355:
4351:
4350:
4348:
4346:
4343:
4342:
4325:
4312:
4301:
4280:
4271:
4264:
4242:
4239:
4198:
4194:
4170:
4166:
4148:
4144:
4124:
4122:
4119:
4118:
4083:
4075:
4064:
4059:
4058:
4050:
4042:
4040:
4037:
4036:
4016:
4015:
4000:
3996:
3987:
3983:
3980:
3975:
3960:
3956:
3947:
3943:
3940:
3925:
3921:
3912:
3908:
3905:
3904:
3899:
3891:
3882:
3878:
3875:
3870:
3865:
3857:
3848:
3844:
3841:
3836:
3828:
3819:
3815:
3811:
3809:
3806:
3805:
3790:
3781:
3773:
3765:
3758:
3752:
3744:
3737:
3731:
3728:
3704:position vector
3696:
3680:
3678:
3675:
3674:
3609:
3606:
3605:
3568:
3565:
3564:
3521:
3518:
3517:
3482:
3477:
3476:
3468:
3465:
3464:
3445:
3443:
3440:
3439:
3419:
3417:
3414:
3413:
3390:
3366:
3340:where Β· is the
3311:
3296:
3292:
3290:
3273:
3267:
3262:
3250:
3230:
3224:
3219:
3213:
3210:
3209:
3188:position vector
3146:
3139:
3135:
3131:
3129:
3110:
3103:
3099:
3095:
3093:
3080:
3073:
3069:
3065:
3063:
3062:
3058:
3045:
3039:
3035:
3033:
3031:
3028:
3027:
3009:
2969:
2965:
2941:
2937:
2919:
2915:
2900:
2896:
2881:
2877:
2868:
2864:
2853:
2851:
2848:
2847:
2811:
2807:
2798:
2794:
2769:
2765:
2756:
2752:
2732:
2728:
2719:
2715:
2713:
2710:
2709:
2703:
2685:has a limit at
2629:
2608:
2605:
2604:
2550:
2547:
2546:
2514:
2491:
2489:
2486:
2485:
2433:
2421:
2418:
2417:
2347:
2344:
2343:
2336:
2320:
2317:
2316:
2310:
2300:
2260:
2257:
2256:
2240:
2237:
2236:
2214:
2182:
2180:
2177:
2176:
2170:
2164:
2144:
2141:
2140:
2114:
2100:
2077:
2074:
2073:
2057:
2055:
2052:
2051:
2020:
1995:
1993:
1990:
1989:
1978:
1966:
1935:
1909:
1904:
1901:
1900:
1881:
1879:
1876:
1875:
1845:
1785:
1782:
1781:
1717:
1714:
1713:
1697:
1695:
1692:
1691:
1681:
1605:
1602:
1601:
1600:, the function
1562:
1559:
1558:
1544:
1526:
1519:
1505:
1453:
1451:
1448:
1447:
1440:
1420:
1392:
1382:
1357:
1354:
1353:
1346:
1340:
1306:
1289:
1286:
1285:
1251:
1234:
1231:
1230:
1213:
1199:
1196:
1195:
1173:
1171:
1168:
1167:
1166:is a subset of
1144:
1130:
1127:
1126:
1099:
1097:
1094:
1093:
1048:
1022:is any integer.
1019:
986:
984:
981:
980:
885:
857:
855:
852:
851:
831:
829:
826:
825:
820:. That is, the
801:
799:
796:
795:
776:
774:
771:
770:
754:
752:
749:
748:
733:
690:
688:
685:
684:
668:
666:
663:
662:
646:
644:
641:
640:
633:complex numbers
612:
610:
607:
606:
595:Euclidean space
574:
572:
569:
568:
529:
527:
524:
523:
503:
501:
498:
497:
455:
419:
380:Binary relation
366:
333:
253:
247:
239:
232:
224:
213:
209:
202:
194:
186:
178:
160:
156:
149:
141:
130:
126:
119:
111:
104:
96:
85:
81:
40:
28:
23:
22:
15:
12:
11:
5:
7410:
7400:
7399:
7394:
7389:
7372:
7371:
7364:
7361:
7360:
7358:
7357:
7352:
7347:
7342:
7337:
7332:
7326:
7325:
7320:
7318:Measure theory
7315:
7312:P-adic numbers
7305:
7300:
7295:
7290:
7285:
7275:
7270:
7264:
7261:
7260:
7258:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7222:
7221:
7220:
7215:
7210:
7200:
7195:
7183:
7180:
7179:
7171:
7170:
7163:
7156:
7148:
7142:
7141:
7133:
7124:
7123:External links
7121:
7120:
7119:
7113:
7098:
7092:
7079:
7073:
7057:
7056:
7049:
7031:
7024:
7005:
7004:
7002:
6999:
6998:
6997:
6992:
6987:
6982:
6975:
6972:
6957:
6952:
6949:
6946:
6942:
6938:
6933:
6929:
6925:
6921:
6918:
6915:
6912:
6889:
6884:
6881:
6878:
6874:
6870:
6865:
6861:
6857:
6853:
6850:
6847:
6844:
6823:
6819:
6815:
6810:
6806:
6785:
6780:
6776:
6772:
6769:
6766:
6763:
6760:
6757:
6753:
6749:
6745:
6741:
6738:
6735:
6714:
6691:
6686:
6681:
6674:
6670:
6665:
6661:
6654:
6650:
6646:
6641:
6637:
6632:
6628:
6621:
6617:
6612:
6604:
6600:
6595:
6591:
6588:
6585:
6579:
6573:
6568:
6564:
6561:
6558:
6555:
6551:
6548:
6545:
6541:
6537:
6532:
6528:
6524:
6520:
6517:
6514:
6511:
6486:
6464:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6436:
6433:
6426:
6423:
6419:
6415:
6411:
6407:
6404:
6401:
6398:
6395:
6391:
6387:
6382:
6378:
6352:
6346:
6341:
6337:
6334:
6328:
6322:
6317:
6313:
6310:
6307:
6304:
6300:
6297:
6291:
6286:
6282:
6278:
6275:
6272:
6269:
6265:
6262:
6256:
6251:
6247:
6243:
6240:
6237:
6234:
6230:
6224:
6219:
6193:
6188:
6167:
6164:
6161:
6157:
6153:
6150:
6147:
6144:
6138:
6133:
6110:
6105:
6102:
6099:
6096:
6092:
6089:
6086:
6083:
6079:
6076:
6056:
6034:
6028:
6023:
6019:
6012:
6007:
6002:
5998:
5993:
5989:
5983:
5978:
5970:
5966:
5961:
5957:
5954:
5948:
5941:
5935:
5930:
5926:
5922:
5918:
5915:
5912:
5909:
5904:
5899:
5895:
5891:
5888:
5885:
5882:
5878:
5875:
5869:
5863:
5858:
5854:
5851:
5848:
5845:
5815:
5810:
5806:
5801:
5797:
5776:
5772:
5768:
5764:
5760:
5757:
5754:
5751:
5745:
5739:
5721:
5718:
5702:
5701:
5639:
5636:
5632:
5631:
5620:
5614:
5611:
5605:
5602:
5596:
5593:
5589:
5584:
5581:
5546:
5543:
5541:, a constant.
5539:speed of light
5511:
5510:
5497:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5472:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5450:
5447:
5444:
5442:
5439:
5432:
5428:
5424:
5421:
5417:
5412:
5405:
5401:
5397:
5394:
5390:
5385:
5382:
5381:
5378:
5375:
5373:
5370:
5363:
5359:
5355:
5352:
5348:
5343:
5340:
5333:
5329:
5325:
5322:
5318:
5313:
5312:
5310:
5305:
5302:
5299:
5296:
5293:
5271:
5270:
5257:
5251:
5248:
5245:
5242:
5240:
5237:
5234:
5231:
5230:
5227:
5224:
5221:
5218:
5215:
5213:
5210:
5207:
5204:
5203:
5201:
5196:
5193:
5190:
5187:
5184:
5161:
5158:
5120:
5069:
4970:
4967:
4954:
4947:
4940:
4929:
4928:
4917:
4914:
4908:
4905:
4900:
4897:
4894:
4890:
4886:
4880:
4877:
4873:
4869:
4865:
4861:
4843:
4842:
4831:
4828:
4822:
4819:
4814:
4811:
4808:
4803:
4799:
4795:
4789:
4784:
4780:
4776:
4771:
4767:
4763:
4760:
4757:
4754:
4748:
4745:
4740:
4737:
4734:
4729:
4725:
4721:
4715:
4710:
4706:
4702:
4697:
4693:
4689:
4686:
4680:
4677:
4672:
4669:
4666:
4661:
4657:
4653:
4647:
4642:
4638:
4634:
4629:
4625:
4621:
4594:
4591:
4590:
4589:
4575:
4572:
4568:
4564:
4561:
4558:
4553:
4549:
4545:
4538:
4534:
4530:
4527:
4524:
4521:
4516:
4512:
4505:
4502:
4499:
4493:
4490:
4486:
4482:
4479:
4476:
4471:
4467:
4463:
4456:
4452:
4448:
4445:
4442:
4439:
4434:
4430:
4423:
4417:
4414:
4410:
4406:
4403:
4400:
4395:
4391:
4387:
4380:
4376:
4372:
4369:
4366:
4363:
4358:
4354:
4321:
4310:
4299:
4276:
4269:
4262:
4238:
4235:
4227:
4226:
4215:
4212:
4209:
4206:
4201:
4197:
4193:
4190:
4187:
4184:
4181:
4178:
4173:
4169:
4165:
4162:
4159:
4156:
4151:
4147:
4143:
4140:
4137:
4134:
4131:
4127:
4108:
4107:
4096:
4093:
4090:
4086:
4082:
4078:
4073:
4067:
4062:
4057:
4053:
4049:
4045:
4030:
4029:
4014:
4011:
4008:
4003:
3999:
3995:
3990:
3986:
3981:
3979:
3976:
3974:
3971:
3968:
3963:
3959:
3955:
3950:
3946:
3941:
3939:
3936:
3933:
3928:
3924:
3920:
3915:
3911:
3907:
3906:
3902:
3898:
3894:
3890:
3885:
3881:
3876:
3874:
3871:
3868:
3864:
3860:
3856:
3851:
3847:
3842:
3839:
3835:
3831:
3827:
3822:
3818:
3814:
3813:
3786:
3777:
3763:
3756:
3742:
3735:
3727:
3724:
3695:
3683:
3671:
3667:
3666:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3599:
3598:
3587:
3584:
3581:
3578:
3575:
3572:
3558:
3557:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3511:
3510:
3499:
3496:
3493:
3490:
3485:
3480:
3475:
3472:
3448:
3422:
3389:
3386:
3365:
3362:
3338:
3337:
3326:
3323:
3317:
3314:
3309:
3306:
3303:
3299:
3295:
3289:
3286:
3283:
3280:
3276:
3270:
3265:
3261:
3257:
3253:
3249:
3246:
3243:
3240:
3237:
3233:
3227:
3222:
3218:
3176:line integrals
3172:
3171:
3159:
3152:
3149:
3142:
3138:
3134:
3128:
3125:
3122:
3116:
3113:
3106:
3102:
3098:
3092:
3086:
3083:
3076:
3072:
3068:
3061:
3057:
3051:
3048:
3042:
3038:
3007:
2998:
2997:
2986:
2983:
2980:
2977:
2972:
2968:
2964:
2961:
2958:
2955:
2952:
2949:
2944:
2940:
2936:
2933:
2930:
2927:
2922:
2918:
2914:
2911:
2908:
2903:
2899:
2895:
2892:
2889:
2884:
2880:
2876:
2871:
2867:
2863:
2860:
2856:
2837:
2836:
2825:
2822:
2819:
2814:
2810:
2806:
2801:
2797:
2793:
2790:
2787:
2783:
2780:
2777:
2772:
2768:
2764:
2759:
2755:
2750:
2746:
2743:
2740:
2735:
2731:
2727:
2722:
2718:
2702:
2699:
2667:
2666:
2655:
2652:
2649:
2646:
2643:
2638:
2635:
2632:
2628:
2624:
2621:
2618:
2615:
2612:
2590:
2589:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2536:
2535:
2524:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2498:
2494:
2471:
2470:
2459:
2456:
2453:
2450:
2447:
2442:
2439:
2436:
2432:
2428:
2425:
2389:of the domain
2385:be a point in
2363:
2360:
2357:
2354:
2351:
2324:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2244:
2224:
2221:
2217:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2185:
2148:
2117:
2113:
2110:
2107:
2103:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2060:
2040:continuous map
2019:
2016:
1998:
1954:
1951:
1948:
1945:
1942:
1938:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1912:
1908:
1884:
1856:
1855:
1833:
1830:
1827:
1824:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1789:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1700:
1662:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1590:
1578:
1575:
1572:
1569:
1566:
1543:
1540:
1521:
1514:
1500:
1472:of a function
1456:
1439:
1436:
1370:
1367:
1364:
1361:
1352:of a function
1342:Main article:
1339:
1336:
1324:
1323:
1310:
1305:
1302:
1299:
1296:
1293:
1283:
1272:
1269:
1266:
1263:
1259:
1254:
1250:
1247:
1244:
1241:
1238:
1228:
1216:
1212:
1209:
1206:
1203:
1176:
1160:
1159:
1147:
1143:
1140:
1137:
1134:
1102:
1047:
1044:
1043:
1042:
1031:
1030:
1023:
1007:
1004:
1001:
998:
993:
990:
973:
958:
957:
950:
947:absolute value
939:
938:
927:
926:
921:
911:
893:differentiable
884:
883:Basic examples
881:
864:
860:
834:
804:
783:
779:
757:
732:
729:
720:of the curve.
693:
671:
649:
631:, such as the
615:
577:
554:real functions
550:differentiable
532:
506:
457:
456:
454:
453:
446:
439:
431:
428:
427:
421:
420:
418:
417:
412:
407:
402:
397:
392:
387:
382:
376:
373:
372:
368:
367:
365:
364:
359:
354:
349:
343:
340:
339:
335:
334:
332:
331:
326:
321:
316:
311:
306:
301:
296:
291:
286:
281:
276:
271:
265:
262:
261:
255:
254:
252:
251:
245:
236:
230:
221:
211:
206:
200:
184:
173:
172:
168:
158:
153:
147:
138:
128:
123:
117:
108:
102:
93:
83:
77:
74:
73:
62:
61:
55:
54:
37:
36:
26:
9:
6:
4:
3:
2:
7409:
7398:
7395:
7393:
7390:
7388:
7385:
7384:
7382:
7369:
7368:
7362:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7336:
7333:
7331:
7328:
7327:
7324:
7321:
7319:
7316:
7313:
7309:
7306:
7304:
7301:
7299:
7296:
7294:
7291:
7289:
7286:
7283:
7279:
7276:
7274:
7271:
7269:
7268:Real analysis
7266:
7265:
7262:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7219:
7216:
7214:
7211:
7209:
7206:
7205:
7204:
7201:
7199:
7196:
7194:
7190:
7189:
7185:
7184:
7181:
7177:
7169:
7164:
7162:
7157:
7155:
7150:
7149:
7146:
7140:
7139:
7134:
7132:
7131:
7127:
7126:
7116:
7114:354-065-340-6
7110:
7106:
7105:
7099:
7095:
7089:
7085:
7080:
7076:
7070:
7066:
7061:
7060:
7052:
7046:
7042:
7035:
7027:
7025:0-471-60840-8
7021:
7017:
7010:
7006:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6980:Real analysis
6978:
6977:
6971:
6950:
6931:
6927:
6882:
6863:
6859:
6808:
6804:
6778:
6774:
6770:
6764:
6758:
6755:
6739:
6736:
6702:
6689:
6679:
6672:
6663:
6659:
6652:
6644:
6639:
6630:
6626:
6619:
6602:
6593:
6586:
6549:
6530:
6526:
6500:
6424:
6421:
6405:
6402:
6396:
6380:
6376:
6368:
6363:
6350:
6339:
6335:
6298:
6284:
6263:
6249:
6228:
6217:
6208:
6186:
6162:
6151:
6148:
6142:
6131:
6103:
6097:
6077:
6074:
6054:
6045:
6032:
6021:
6017:
6005:
6000:
5991:
5987:
5968:
5959:
5952:
5933:
5876:
5834:
5832:
5808:
5804:
5799:
5795:
5758:
5755:
5749:
5727:
5717:
5714:
5708:
5697:
5693:
5686:
5682:
5675:
5671:
5667:
5666:
5665:
5661:
5657:
5651:
5649:
5645:
5635:
5609:
5603:
5594:
5579:
5572:
5571:
5570:
5568:
5564:
5560:
5556:
5555:Hilbert space
5552:
5542:
5540:
5536:
5532:
5528:
5524:
5520:
5516:
5495:
5489:
5484:
5479:
5474:
5467:
5462:
5457:
5452:
5445:
5440:
5430:
5426:
5422:
5419:
5415:
5403:
5399:
5395:
5392:
5388:
5383:
5376:
5371:
5361:
5357:
5353:
5350:
5346:
5341:
5331:
5327:
5323:
5320:
5316:
5308:
5303:
5297:
5284:
5283:
5282:
5280:
5276:
5255:
5249:
5246:
5243:
5238:
5235:
5232:
5225:
5222:
5219:
5216:
5211:
5208:
5205:
5199:
5194:
5188:
5182:
5175:
5174:
5173:
5171:
5167:
5157:
5155:
5151:
5147:
5143:
5139:
5136:
5131:
5127:
5123:
5118:
5114:
5110:
5107:
5102:
5098:
5094:
5090:
5086:
5082:
5076:
5072:
5067:
5063:
5059:
5056:
5051:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
5000:
4992:
4988:
4984:
4980:
4975:
4966:
4964:
4957:
4953:
4946:
4939:
4935:
4915:
4912:
4906:
4903:
4895:
4884:
4878:
4867:
4852:
4851:
4850:
4848:
4829:
4826:
4820:
4817:
4809:
4801:
4797:
4793:
4782:
4778:
4774:
4769:
4765:
4758:
4755:
4752:
4746:
4743:
4735:
4727:
4723:
4719:
4708:
4704:
4700:
4695:
4691:
4684:
4678:
4675:
4667:
4659:
4655:
4651:
4640:
4636:
4632:
4627:
4623:
4612:
4611:
4610:
4608:
4604:
4600:
4573:
4570:
4566:
4559:
4551:
4547:
4543:
4536:
4532:
4528:
4522:
4514:
4510:
4503:
4500:
4497:
4491:
4488:
4484:
4477:
4469:
4465:
4461:
4454:
4450:
4446:
4440:
4432:
4428:
4421:
4415:
4412:
4408:
4401:
4393:
4389:
4385:
4378:
4374:
4370:
4364:
4356:
4352:
4341:
4340:
4339:
4337:
4333:
4329:
4324:
4320:
4316:
4309:
4305:
4298:
4294:
4290:
4286:
4279:
4275:
4268:
4261:
4257:
4253:
4249:
4245:
4234:
4232:
4207:
4199:
4195:
4191:
4188:
4185:
4179:
4171:
4167:
4163:
4157:
4149:
4145:
4138:
4132:
4117:
4116:
4115:
4113:
4091:
4080:
4071:
4065:
4047:
4035:
4034:
4033:
4009:
4001:
3997:
3993:
3988:
3984:
3977:
3969:
3961:
3957:
3953:
3948:
3944:
3934:
3926:
3922:
3918:
3913:
3909:
3888:
3883:
3879:
3872:
3854:
3849:
3845:
3825:
3820:
3816:
3804:
3803:
3802:
3800:
3794:
3789:
3785:
3780:
3776:
3769:
3762:
3755:
3748:
3741:
3734:
3720:
3716:
3712:
3708:
3705:
3700:
3670:
3653:
3650:
3644:
3638:
3635:
3632:
3629:
3623:
3620:
3617:
3611:
3604:
3603:
3602:
3582:
3576:
3573:
3570:
3563:
3562:
3561:
3544:
3541:
3535:
3532:
3529:
3523:
3516:
3515:
3514:
3494:
3483:
3473:
3470:
3463:
3462:
3461:
3437:
3411:
3407:
3403:
3399:
3397:
3385:
3383:
3379:
3375:
3371:
3361:
3359:
3355:
3351:
3347:
3343:
3324:
3321:
3315:
3312:
3304:
3293:
3287:
3281:
3268:
3263:
3259:
3255:
3247:
3244:
3238:
3225:
3220:
3216:
3208:
3207:
3206:
3204:
3200:
3196:
3192:
3189:
3185:
3181:
3177:
3157:
3150:
3147:
3140:
3136:
3132:
3126:
3123:
3120:
3114:
3111:
3104:
3100:
3096:
3090:
3084:
3081:
3074:
3070:
3066:
3059:
3055:
3049:
3046:
3036:
3026:
3025:
3024:
3022:
3019:= 1, 2, ...,
3018:
3014:
3010:
3003:
2978:
2970:
2966:
2962:
2959:
2956:
2950:
2942:
2938:
2934:
2928:
2920:
2916:
2909:
2901:
2897:
2893:
2890:
2887:
2882:
2878:
2874:
2869:
2865:
2858:
2846:
2845:
2844:
2842:
2820:
2812:
2808:
2804:
2799:
2795:
2791:
2788:
2785:
2778:
2770:
2766:
2762:
2757:
2753:
2748:
2741:
2733:
2729:
2725:
2720:
2716:
2708:
2707:
2706:
2698:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2653:
2647:
2641:
2636:
2630:
2622:
2616:
2610:
2603:
2602:
2601:
2599:
2595:
2576:
2573:
2570:
2564:
2561:
2558:
2552:
2545:
2544:
2543:
2541:
2522:
2519:
2511:
2508:
2502:
2496:
2484:
2483:
2482:
2480:
2476:
2457:
2451:
2445:
2440:
2434:
2426:
2423:
2416:
2415:
2414:
2412:
2409:tends toward
2408:
2404:
2400:
2396:
2392:
2388:
2384:
2380:
2375:
2361:
2355:
2349:
2340:
2322:
2313:
2308:
2303:
2286:
2283:
2280:
2274:
2271:
2268:
2262:
2242:
2222:
2219:
2208:
2202:
2199:
2193:
2187:
2173:
2167:
2162:
2146:
2138:
2134:
2129:
2111:
2108:
2105:
2097:
2091:
2088:
2085:
2079:
2049:
2044:
2041:
2037:
2033:
2024:
2015:
2013:
1985:
1981:
1976:
1970:
1952:
1946:
1940:
1936:
1932:
1923:
1917:
1914:
1910:
1906:
1897:
1873:
1869:
1868:neighbourhood
1865:
1861:
1852:
1848:
1828:
1822:
1815:
1809:
1800:
1794:
1791:
1787:
1764:
1758:
1755:
1749:
1743:
1734:
1728:
1725:
1722:
1719:
1688:
1684:
1679:
1675:
1671:
1667:
1663:
1660:
1656:
1637:
1631:
1628:
1619:
1613:
1610:
1607:
1599:
1595:
1591:
1576:
1567:
1557:
1553:
1549:
1548:
1547:
1539:
1537:
1533:
1528:
1525:
1518:
1513:
1509:
1504:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1471:
1445:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1406:
1404:
1400:
1395:
1390:
1385:
1365:
1359:
1351:
1345:
1335:
1333:
1329:
1326:which is the
1308:
1303:
1297:
1291:
1284:
1267:
1264:
1261:
1257:
1248:
1245:
1239:
1236:
1229:
1207:
1204:
1201:
1194:
1193:
1192:
1189:
1165:
1138:
1135:
1132:
1125:
1124:
1123:
1121:
1117:
1091:
1086:
1084:
1080:
1076:
1072:
1068:
1065:
1061:
1057:
1053:
1040:
1036:
1035:
1034:
1028:
1024:
1005:
1002:
999:
996:
991:
988:
978:
974:
971:
967:
963:
962:
961:
955:
951:
948:
944:
943:
942:
936:
932:
931:
930:
925:
922:
919:
915:
912:
910:
906:
902:
898:
897:
896:
894:
890:
880:
878:
862:
849:
823:
819:
781:
746:
737:
731:Real function
728:
726:
721:
719:
715:
711:
706:
638:
634:
630:
604:
601:, the set of
600:
596:
592:
565:
563:
559:
555:
551:
547:
521:
496:
492:
488:
484:
480:
476:
472:
468:
464:
452:
447:
445:
440:
438:
433:
432:
430:
429:
426:
423:
422:
416:
413:
411:
408:
406:
403:
401:
398:
396:
393:
391:
388:
386:
383:
381:
378:
377:
375:
374:
370:
369:
363:
360:
358:
355:
353:
350:
348:
345:
344:
342:
341:
338:Constructions
337:
336:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
272:
270:
267:
266:
264:
263:
260:
257:
256:
250:
237:
235:
222:
220:
207:
205:
192:
191:
190:
189:
175:
171:
169:
167:
154:
152:
139:
137:
124:
122:
109:
107:
94:
92:
79:
78:
76:
75:
72:
68:
64:
63:
60:
57:
56:
51:
47:
43:
39:
38:
35:
32:
31:
19:
18:Real function
7392:Real numbers
7365:
7186:
7137:
7129:
7107:. Springer.
7103:
7083:
7064:
7050:0-07-054235X
7040:
7034:
7015:
7014:R. Courant.
7009:
6703:
6501:
6364:
6209:
6046:
5835:
5723:
5712:
5706:
5703:
5695:
5691:
5684:
5680:
5673:
5669:
5659:
5655:
5652:
5643:
5641:
5633:
5551:Banach space
5548:
5534:
5526:
5522:
5518:
5514:
5512:
5272:
5163:
5150:acceleration
5145:
5141:
5137:
5134:
5132:
5125:
5121:
5116:
5112:
5108:
5105:
5100:
5096:
5092:
5088:
5084:
5080:
5074:
5070:
5065:
5061:
5057:
5054:
5049:
5045:
5041:
5037:
5029:
5028:), treating
5025:
5021:
5009:
5005:
5001:
4998:
4996:
4990:
4986:
4982:
4978:
4963:in the plane
4962:
4955:
4951:
4944:
4937:
4933:
4930:
4844:
4606:
4602:
4598:
4596:
4335:
4331:
4327:
4322:
4318:
4314:
4307:
4303:
4296:
4288:
4284:
4277:
4273:
4266:
4259:
4255:
4251:
4247:
4243:
4240:
4228:
4111:
4109:
4031:
3798:
3792:
3787:
3783:
3778:
3774:
3767:
3760:
3753:
3746:
3739:
3732:
3729:
3718:
3714:
3710:
3706:
3668:
3600:
3559:
3512:
3436:zero element
3409:
3405:
3401:
3394:real-valued
3393:
3391:
3367:
3357:
3353:
3349:
3345:
3339:
3202:
3198:
3194:
3190:
3183:
3173:
3020:
3016:
3012:
3005:
3001:
2999:
2840:
2838:
2704:
2694:
2690:
2686:
2682:
2678:
2670:
2668:
2597:
2593:
2591:
2539:
2537:
2478:
2474:
2472:
2410:
2406:
2402:
2401:has a limit
2398:
2394:
2390:
2382:
2376:
2342:centered at
2338:
2315:centered at
2311:
2306:
2301:
2171:
2165:
2136:
2132:
2130:
2045:
2029:
1983:
1979:
1974:
1968:
1898:
1896:-algebras).
1863:
1859:
1857:
1850:
1846:
1686:
1682:
1677:
1673:
1669:
1665:
1658:
1654:
1597:
1593:
1551:
1545:
1529:
1523:
1516:
1511:
1507:
1502:
1497:
1493:
1489:
1485:
1481:
1477:
1476:to a subset
1473:
1469:
1441:
1429:
1425:
1421:
1413:
1407:
1393:
1388:
1383:
1347:
1331:
1325:
1190:
1163:
1161:
1089:
1087:
1082:
1078:
1074:
1070:
1066:
1051:
1049:
1032:
959:
940:
928:
903:, including
886:
818:real numbers
742:
722:
707:
591:vector space
566:
553:
495:real numbers
482:
460:
405:Higher-order
177:
176:
174:
170:
49:
45:
41:
7193:Integration
5726:cardinality
5525:, in which
5133:Similarly,
4985:, velocity
4981:, position
4961:are points
4847:dot product
4241:At a point
4231:space curve
3801:, so that:
3726:Formulation
3342:dot product
3180:space curve
2413:, denoted
2139:at a point
2131:A function
1486:restriction
1328:square root
1060:real number
1039:square root
637:quaternions
475:engineering
390:Multivalued
352:Composition
347:Restriction
7381:Categories
7218:stochastic
7001:References
2673:is in the
2255:such that
2175:such that
2137:continuous
1977:such that
1680:such that
1532:continuity
954:cubic root
889:continuous
846:, and its
324:Surjective
314:Measurable
309:Continuous
284:Polynomial
7330:Functions
6883:≥
6771:≡
6748:→
6669:ℵ
6649:ℵ
6645:⋅
6636:ℵ
6616:ℵ
6599:ℵ
6550:≤
6414:→
6299:≤
6264:≤
6160:→
6104:≤
6078:≤
6006:⋅
5997:ℵ
5965:ℵ
5831:continuum
5796:ℶ
5767:→
5619:Ψ
5613:^
5601:Ψ
5592:∂
5588:∂
5583:ℏ
5427:β
5423:−
5400:β
5396:−
5389:β
5384:−
5358:β
5354:−
5347:β
5342:−
5328:β
5324:−
5298:β
5292:Λ
5250:θ
5247:
5239:θ
5236:
5226:θ
5223:
5217:−
5212:θ
5209:
5189:θ
4879:⋅
4868:−
4775:−
4756:⋯
4701:−
4633:−
4529:−
4501:⋯
4447:−
4371:−
4189:…
4056:→
3978:⋯
3897:→
3873:⋯
3863:→
3834:→
3636:−
3612:ϕ
3524:ϕ
3489:→
3471:ϕ
3288:⋅
3260:∫
3245:⋅
3217:∫
3124:…
2960:…
2891:…
2789:…
2681:, and if
2634:→
2574:δ
2523:ε
2509:−
2438:→
2284:φ
2223:ε
2200:−
2159:which is
2109:−
1930:↦
1807:↦
1741:↦
1626:↦
1574:↦
1265:≥
1249:∈
1211:→
1142:→
1003:π
989:π
920:functions
329:Bijective
319:Injective
294:Algebraic
65:Types by
7355:Infinity
7208:ordinary
7188:Calculus
7065:Calculus
6974:See also
6207:, since
6178:is also
5650:values.
5563:operator
5148:is the "
5018:particle
5014:velocity
5012:is the "
4317:), ...,
4114:-tuple,
3364:Theorems
3178:along a
2701:Calculus
2675:boundary
2538:for all
2235:for all
2161:interior
1418:equation
1410:preimage
1399:interval
1120:interval
1083:function
1064:variable
1056:function
877:interval
848:codomain
745:function
603:matrices
562:codomain
546:interval
487:function
467:geometry
410:Morphism
395:Implicit
299:Analytic
289:Rational
274:Identity
269:Constant
71:codomain
48: (
34:Function
7213:partial
5648:complex
5537:is the
5529:is the
5172:in 2d:
4950:, ...,
4330:), and
4272:, ...,
3772:, ...,
3434:to the
3186:, with
1712:, then
635:or the
629:algebra
518:, or a
493:is the
415:Functor
385:Partial
362:Inverse
7350:Series
7111:
7090:
7071:
7047:
7022:
6428:
5704:where
5561:or an
5277:, the
5166:matrix
4931:where
3376:, and
3344:, and
3015:) for
2038:and a
2012:fields
1554:, the
1534:or by
1444:domain
1438:Domain
1116:domain
1114:, the
1018:where
918:cosine
822:domain
794:where
520:subset
491:domain
489:whose
477:, and
304:Smooth
279:Linear
67:domain
7345:Limit
5787:, is
5553:or a
3713:. At
2669:When
2405:when
2379:limit
1986:) β 0
1661:= 0).
1350:image
1338:Image
1071:value
1054:is a
970:zeros
714:curve
710:image
485:is a
400:Space
7109:ISBN
7088:ISBN
7069:ISBN
7045:ISBN
7020:ISBN
5724:The
5710:and
4609:is:
4254:) =
3513:and
3352:and
2571:<
2520:<
2377:The
2281:<
2220:<
1780:and
1676:and
1668:and
1510:and
1442:The
1408:The
1348:The
1037:The
1025:The
975:The
952:The
945:The
933:The
916:and
914:Sine
907:and
899:All
891:and
708:The
597:, a
481:, a
69:and
5653:If
5559:ket
5244:cos
5233:sin
5220:sin
5206:cos
4936:= (
4306:),
4295:of
4258:= (
3438:in
2693:to
2627:lim
2431:lim
2135:is
2050:of
1664:If
1492:to
1488:of
1330:of
1092:of
850:is
769:to
522:of
461:In
7383::
7191::
6970:.
6834:,
5692:ih
5689:+
5678:=
5642:A
5569::
5517:=
5164:A
5156:.
5146:dt
5144:)/
5130:.
5124:=
5117:dt
5115:)/
5099:β
5091:=
5083:=
5073:=
5066:dt
5064:)/
5048:=
5010:dt
5008:)/
4943:,
4849::
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4338::
4287:=
4265:,
4250:=
4233:.
3782:=
3759:=
3751:,
3738:=
3717:=
3404:=
3392:A
3384:.
3372:,
3356:=
3348:=
3205::
3193:=
3023::
2843::
2697:.
2014:.
1527:.
1480:β
1434:.
1424:=
1405:.
1334:.
1050:A
964:A
473:,
469:,
243:β
228:β
215:β
198:β
182:β
162:β
145:β
132:β
115:β
113:πΉ
100:β
98:πΉ
89:πΉ
87:β
44:β¦
7314:)
7310:(
7284:)
7280:(
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7096:.
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6956:c
6951:=
6948:)
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6784:}
6779:0
6775:x
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6765:x
6762:(
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6744:R
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6734:{
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6690:.
6685:c
6680:=
6673:0
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6603:0
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6587:=
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6578:Q
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6485:c
6463:}
6459:s
6456:u
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6425:f
6422::
6418:R
6410:R
6406::
6403:f
6400:{
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6390:R
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6340:2
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6316:(
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6290:R
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6268:c
6261:)
6255:R
6250:2
6246:(
6242:d
6239:r
6236:a
6233:c
6229:=
6223:c
6218:2
6192:c
6187:2
6166:}
6163:X
6156:R
6152::
6149:f
6146:{
6143:=
6137:R
6132:X
6109:c
6101:)
6098:X
6095:(
6091:d
6088:r
6085:a
6082:c
6075:2
6055:X
6033:.
6027:c
6022:2
6018:=
6011:c
6001:0
5992:2
5988:=
5982:c
5977:)
5969:0
5960:2
5956:(
5953:=
5947:c
5940:c
5934:=
5929:)
5925:R
5921:(
5917:d
5914:r
5911:a
5908:c
5903:)
5898:R
5894:(
5890:d
5887:r
5884:a
5881:c
5877:=
5874:)
5868:R
5862:R
5857:(
5853:d
5850:r
5847:a
5844:c
5814:c
5809:2
5805:=
5800:2
5775:}
5771:R
5763:R
5759::
5756:f
5753:{
5750:=
5744:R
5738:R
5713:h
5707:g
5700:,
5698:)
5696:x
5694:(
5687:)
5685:x
5683:(
5681:g
5676:)
5674:x
5672:(
5670:f
5662:)
5660:x
5658:(
5656:f
5610:H
5604:=
5595:t
5580:i
5535:c
5527:v
5523:c
5521:/
5519:v
5515:Ξ²
5496:]
5490:1
5485:0
5480:0
5475:0
5468:0
5463:1
5458:0
5453:0
5446:0
5441:0
5431:2
5420:1
5416:1
5404:2
5393:1
5377:0
5372:0
5362:2
5351:1
5332:2
5321:1
5317:1
5309:[
5304:=
5301:)
5295:(
5256:]
5200:[
5195:=
5192:)
5186:(
5183:R
5142:t
5140:(
5138:r
5135:d
5126:c
5122:t
5119:|
5113:t
5111:(
5109:r
5106:d
5101:a
5097:p
5093:c
5089:t
5085:c
5081:t
5075:c
5071:t
5068:|
5062:t
5060:(
5058:r
5055:d
5050:c
5046:t
5042:t
5038:t
5030:r
5026:t
5024:(
5022:r
5006:t
5004:(
5002:r
4999:d
4993:.
4991:a
4987:v
4983:r
4979:m
4959:)
4956:n
4952:p
4948:2
4945:p
4941:1
4938:p
4934:p
4916:0
4913:=
4907:t
4904:d
4899:)
4896:t
4893:(
4889:r
4885:d
4876:)
4872:a
4864:p
4860:(
4830:0
4827:=
4821:t
4818:d
4813:)
4810:t
4807:(
4802:n
4798:r
4794:d
4788:)
4783:n
4779:a
4770:n
4766:p
4762:(
4759:+
4753:+
4747:t
4744:d
4739:)
4736:t
4733:(
4728:2
4724:r
4720:d
4714:)
4709:2
4705:a
4696:2
4692:p
4688:(
4685:+
4679:t
4676:d
4671:)
4668:t
4665:(
4660:1
4656:r
4652:d
4646:)
4641:1
4637:a
4628:1
4624:p
4620:(
4607:a
4603:r
4599:n
4574:t
4571:d
4567:/
4563:)
4560:t
4557:(
4552:n
4548:r
4544:d
4537:n
4533:a
4526:)
4523:t
4520:(
4515:n
4511:r
4504:=
4498:=
4492:t
4489:d
4485:/
4481:)
4478:t
4475:(
4470:2
4466:r
4462:d
4455:2
4451:a
4444:)
4441:t
4438:(
4433:2
4429:r
4422:=
4416:t
4413:d
4409:/
4405:)
4402:t
4399:(
4394:1
4390:r
4386:d
4379:1
4375:a
4368:)
4365:t
4362:(
4357:1
4353:r
4336:t
4332:r
4328:t
4326:(
4323:n
4319:r
4315:t
4313:(
4311:2
4308:r
4304:t
4302:(
4300:1
4297:r
4289:c
4285:t
4281:)
4278:n
4274:a
4270:2
4267:a
4263:1
4260:a
4256:a
4252:c
4248:t
4246:(
4244:r
4214:]
4211:)
4208:t
4205:(
4200:n
4196:r
4192:,
4186:,
4183:)
4180:t
4177:(
4172:2
4168:r
4164:,
4161:)
4158:t
4155:(
4150:1
4146:r
4142:[
4139:=
4136:)
4133:t
4130:(
4126:r
4112:n
4095:)
4092:t
4089:(
4085:r
4081:=
4077:r
4072:,
4066:n
4061:R
4052:R
4048::
4044:r
4013:)
4010:t
4007:(
4002:n
3998:r
3994:=
3989:n
3985:r
3973:)
3970:t
3967:(
3962:2
3958:r
3954:=
3949:2
3945:r
3938:)
3935:t
3932:(
3927:1
3923:r
3919:=
3914:1
3910:r
3901:R
3893:R
3889::
3884:n
3880:r
3867:R
3859:R
3855::
3850:2
3846:r
3838:R
3830:R
3826::
3821:1
3817:r
3799:t
3795:)
3793:t
3791:(
3788:n
3784:r
3779:n
3775:r
3770:)
3768:t
3766:(
3764:2
3761:r
3757:2
3754:r
3749:)
3747:t
3745:(
3743:1
3740:r
3736:1
3733:r
3719:a
3715:r
3711:t
3707:r
3682:R
3654:0
3651:=
3648:)
3645:x
3642:(
3639:f
3633:y
3630:=
3627:)
3624:y
3621:,
3618:x
3615:(
3586:)
3583:x
3580:(
3577:f
3574:=
3571:y
3545:0
3542:=
3539:)
3536:y
3533:,
3530:x
3527:(
3498:}
3495:0
3492:{
3484:2
3479:R
3474::
3447:R
3421:R
3410:x
3408:(
3406:f
3402:y
3358:b
3354:x
3350:a
3346:x
3325:x
3322:d
3316:x
3313:d
3308:)
3305:x
3302:(
3298:r
3294:d
3285:)
3282:x
3279:(
3275:y
3269:b
3264:a
3256:=
3252:r
3248:d
3242:)
3239:x
3236:(
3232:y
3226:b
3221:a
3203:x
3199:x
3197:(
3195:r
3191:r
3184:x
3158:)
3151:x
3148:d
3141:n
3137:y
3133:d
3127:,
3121:,
3115:x
3112:d
3105:2
3101:y
3097:d
3091:,
3085:x
3082:d
3075:1
3071:y
3067:d
3060:(
3056:=
3050:x
3047:d
3041:y
3037:d
3021:n
3017:i
3013:x
3011:(
3008:i
3006:f
3002:y
2985:]
2982:)
2979:x
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2971:n
2967:f
2963:,
2957:,
2954:)
2951:x
2948:(
2943:2
2939:f
2935:,
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2929:x
2926:(
2921:1
2917:f
2913:[
2910:=
2907:)
2902:n
2898:y
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2883:2
2879:y
2875:,
2870:1
2866:y
2862:(
2859:=
2855:y
2841:x
2824:)
2821:x
2818:(
2813:n
2809:f
2805:=
2800:n
2796:y
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2779:x
2776:(
2771:2
2767:f
2763:=
2758:2
2754:y
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2739:(
2734:1
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2726:=
2721:1
2717:y
2695:a
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2651:)
2648:x
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2642:f
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2623:=
2620:)
2617:a
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2611:f
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2577:.
2568:)
2565:a
2562:,
2559:x
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2553:d
2540:x
2516:|
2512:L
2506:)
2503:x
2500:(
2497:f
2493:|
2479:Ξ΄
2475:Ξ΅
2458:,
2455:)
2452:x
2449:(
2446:f
2441:a
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2427:=
2424:L
2411:a
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2399:f
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2383:a
2362:.
2359:)
2356:a
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2350:f
2339:Ξ΅
2337:2
2323:a
2312:Ο
2307:f
2302:Ο
2287:.
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2263:d
2243:x
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2203:f
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2172:Ο
2166:Ξ΅
2147:a
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2098:=
2095:)
2092:y
2089:,
2086:x
2083:(
2080:d
2059:R
1997:R
1984:x
1982:(
1980:f
1975:f
1971:)
1969:x
1967:(
1953:,
1950:)
1947:x
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1941:f
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1933:1
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1918::
1915:f
1911:/
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1851:Y
1849:β©
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1823:g
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