Continuity at a point

Continuity at a point
Tag	`0EEE2CZB2A`
URI	`m2://0EEE2CZB2A`
Kind	definition
Writing	W0 stub
MSC	26A15
Defines	Continuity at a point
Uses	Limit of a function

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Definition

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ be a function, and let $a \in D$ . The function $f$ is continuous at $a$ if the following holds. For each open interval $V$ containing $f(a)$ there is an open interval $U$ containing $a$ with $f(x) \in V$ for all $x \in U \cap D$ .

Remarks

If $a$ is a limit point of $D$ , then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$ . See limit of a function. If $a$ is an isolated point of $D$ , then each function on $D$ is continuous at $a$ . The epsilon-delta form states the same condition with numbers. Mathlib defines it for topological spaces, as ContinuousAt.

Explanations

L1 human

A function is continuous at $a$ if small changes of $x$ near $a$ give small changes of $f(x)$ . The graph has no jump or gap at $a$ .

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What depends on this

141 nodes use this node, directly or through other nodes.

Power series are differentiable inside the radius of convergence 002X7EK91H
A function with zero derivative is constant 003QDZ4T16
Natural logarithm 0044CPHJBV
A differentiable function is convex iff its derivative is increasing 005JT76HCK
Taylor's theorem with the Cauchy remainder 005Q84BMAD
Euler–Mascheroni constant 00C5P9JZBK
The logarithm grows slower than any power 00DGYY10FV
Intermediate value theorem 00KAFBR32E
Derivative of sine 00R4MWW532
A convex function on an open interval is continuous 011HV2GAWF
1 + x is at most exp x 012C7GFF20
Reduction formula 012RYET5E2
Rolle's theorem 0194DY35BG
Weierstrass function 01A0MK1730
First derivative test 01A2EGXJEM
cos pi = -1 01A8JGR4JM
sin x / x tends to 1 01QSSKS5C1
Taylor series 0225QPWRRE
A function with nonnegative derivative is increasing 026BJBQNGJ
Uniformly continuous functions map Cauchy sequences to Cauchy sequences 026FDZFRXS
Angle addition formulas 028XEQCE88
Inverse trigonometric functions 02HTDK1FBD
Power series of sine 02KTRHQMHN
Cauchy mean value theorem 02SYZAWM2K
Leibniz formula for pi 037FD5NBRT
Integral of exp(-x) over the half line 039JAF7ZAN
Pythagorean trigonometric identity 03H7WZ4T3K
The Dirichlet function is not Riemann integrable 03Z4JC4YRJ
Maclaurin series 043SJ5RF59
log x is at most x - 1 04CBRA4JRK
Inverse hyperbolic functions 04D7CHT5V1
Classification of discontinuities 04R9K5H562
Integral of cosine 04WTBQ7APQ
Lebesgue's criterion for Riemann integrability 0565EGK6XP
Derivative of arctangent 05P4B4Q22V
Harmonic numbers grow like log n 05S2GSPMGA
First fundamental theorem of calculus 063H5KJC69
Arctangent 065XW2V6C3
Integral of the exponential function 06D0EWR2DV
Power series of the exponential function 06JZGMYTYD
Continuous functions are Riemann integrable 06SESV7GQY
Continuously differentiable function 06SKCHDQT5
Tangent function 06T0HS14PP
Continuous image of a closed interval is a closed interval 06ZARTQTQE
Continuity of the inverse of a strictly monotone continuous function 07BY47HG5S
A function with positive derivative is strictly increasing 07J5X2T1ZX
Derivatives of the hyperbolic functions 07T2TKGZ4J
Derivative of the logarithm 07X1QA6BRW
Differentiable functions are continuous 07ZMX4ANJS
Power series of cosine 08W5RN375X
Integration by parts 08Y1BBQTK9
Smooth non-analytic function 09RTYYHXKA
Uniform continuity on a set 0A3CK6DVVB
x^2 is not uniformly continuous on the real line 0A8JXHYTJC
Boundedness theorem 0ATVXC3NCC
Arcsine 0B89D93YA1
Mean value theorem 0BARQY52N5
Smooth function 0BE3QDH73A
Quotient rule 0BS0NJPKVE
pi 0C1VYVNV2N
Polynomial functions are continuous 0C5TP6RYSV
Absolute value is continuous but not differentiable at 0 0CQ9ZM383N
Uniform convergence of derivatives allows term-by-term differentiation 0CS42KEM5C
Integration by substitution 0D37QFPEZE
A continuous injective function on an interval is strictly monotone 0D6FCW80A0
The exponential function is convex 0D7GRZ2ADX
Basel problem 0DMY75D6FV
Trigonometric substitution 0EAR3KNRTN
x sin(1/x) is continuous at 0 0EZMAY1FQD
The gamma function extends the factorial 0FDCW77H44
Second derivative test 0FZFHK97P7
Derivative of arcsine 0GA6G43EA9
Quotient of continuous functions is continuous 0GR2APWTAC
Area under a curve 0GXA8Z205E
Integral of sine 0GXCXCX3QN
Dirichlet integral 0GZNSHMG4Q
Taylor's theorem with the Lagrange remainder 0H5ZAZCX2T
Thomae's function 0HJ28H9Y95
Existence of nth roots by the intermediate value theorem 0J29GM30Z7
Logarithm of a quotient 0J7PHRBNNS
Weierstrass approximation theorem 0JQE130DHW
L'Hôpital's rule 0JWT4PPQ7B
Power rule 0JZGW75C5Z
Lipschitz functions are uniformly continuous 0K1EWAR6AF
Integral of 1/x 0K1N1Y4EGZ
Logarithm of a product 0K2FNY6DYV
General Leibniz rule 0K9QCK6WJC
Stirling's formula 0KC0F51T6A
The exponential function is strictly increasing 0KN2V7840E
Derivative of x^a 0KY08TWDMQ
Derivative of an inverse function 0M0RZ3HSB0
Derivative of a polynomial 0MH1WK89PP
Mean value theorem for integrals 0N91N98EG5
The integral of a continuous function is an antiderivative 0NC2A3HWGT
Fundamental theorem of calculus 0NHGY0N0JB
Composition of continuous functions is continuous 0NRQPQX3VA
Continuous function 0P27W9RW0G
Sine is 2π-periodic 0P4NQD746K
A function with bounded derivative is Lipschitz 0P64VE79X3
Continuity is equivalent to sequential continuity 0PAQM3H57X
Integral of a power 0Q3HYFT9RQ
Uniform continuity 0Q4DR37DVE
Darboux's theorem 0Q5A47J48D
Reciprocal rule 0QD4E5JYPT
Removable discontinuity 0QG9C579SR
Irrationality of pi 0QHZX1PK8V
Continuity of the sum of a uniformly convergent series of functions 0R0MYT0BJD
pi is greater than 3 0R7SVNYTJS
Arccosine 0RDP2HEFEW
Logarithm is the inverse of the exponential 0RKW2X63YR
A continuous nowhere differentiable function 0RXK2Z808Q
Chain rule 0RXX2PT1MQ
Power-reduction formula for cosine 0S049HG1W5
Indeterminate form 0S7G92KKA7
Derivative of tangent 0SAMR34WXC
Power series of arctangent 0T32ME0A6H
Dirichlet function 0T4EJ8ZJM9
Term-by-term differentiation 0TJ8QBJX5K
Real exponentiation 0TV07AQT62
Derivative of cosine 0V2DV29EGT
Square root function 0V7E9KP6Q2
Logarithm of a power 0V948DE5KA
Product rule 0V9A43H7WV
Improper integral of x^p on an unbounded interval 0VGKB2NC4T
Product of continuous functions is continuous 0VGMHCF4SC
Continuous functions on a closed bounded interval are uniformly continuous 0W0TZDRF9C
Tangent half-angle substitution 0W8ETY2AZ6
Epsilon-delta definition of continuity 0W8YHHGCJG
Sine and cosine 0WB57D9488
Integration by partial fractions 0WBTWFXJQG
Rational function 0WCZH4AFMT
x^2 sin(1/x) is differentiable with discontinuous derivative 0WDAKKN4NC
Taylor's theorem with the integral remainder 0WJGR81HBD
Extreme value theorem 0WS1A8E2P9
Uniform limit theorem 0X9XT05JVC
Wallis product 0XCYA4YERS
Second fundamental theorem of calculus 0XYAFCN0PT
x^n converges pointwise but not uniformly on [0, 1] 0YSRR1JTKZ
Mercator series 0YZXNE3HFR
A function with nonnegative second derivative is convex 0Z41C17TGP
The logarithm is strictly concave 0ZS54B9GJF

Sources

Web https://en.wikipedia.org/wiki/Continuous_function, Definition in terms of neighborhoods

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Used by

Classification of discontinuities 04R9K5H562
Lebesgue's criterion for Riemann integrability 0565EGK6XP
Differentiable functions are continuous 07ZMX4ANJS
x sin(1/x) is continuous at 0 0EZMAY1FQD
Thomae's function 0HJ28H9Y95
Continuous function 0P27W9RW0G
Removable discontinuity 0QG9C579SR
Dirichlet function 0T4EJ8ZJM9
Epsilon-delta definition of continuity 0W8YHHGCJG
Uniform limit theorem 0X9XT05JVC