Definition in terms of limits of functions

The function f is continuous at some point c of its domain if the limit of ${\displaystyle f(x),}$ as x approaches c through the domain of f, exists and is equal to ${\displaystyle f(c).}$^[9] In mathematical notation, this is written as

$${\displaystyle \lim _{x\to c}{f(x)}=f(c).}$$

In detail this means three conditions: first, f has to be defined at c (guaranteed by the requirement that c is in the domain of f). Second, the limit of that equation has to exist. Third, the value of this limit must equal ${\displaystyle f(c).}$

(Here, we have assumed that the domain of f does not have any isolated points.)

Definition in terms of neighborhoods

A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point ${\displaystyle f(c)}$ as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood ${\displaystyle N_{1}(f(c))}$ there is a neighborhood ${\displaystyle N_{2}(c)}$ in its domain such that ${\displaystyle f(x)\in N_{1}(f(c))}$ whenever ${\displaystyle x\in N_{2}(c).}$

As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous.

Definition in terms of limits of sequences

The sequence exp(1/n) converges to exp(0) = 1

One can instead require that for any sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ of points in the domain which converges to c, the corresponding sequence ${\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}$ converges to ${\displaystyle f(c).}$ In mathematical notation,

$${\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}$$

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Illustration of the ε-δ-definition: at x = 2, any value δ ≤ 0.5 satisfies the condition of the definition for ε = 0.5.

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function ${\displaystyle f:D\to \mathbb {R} }$ as above and an element ${\displaystyle x_{0}}$ of the domain ${\displaystyle D}$, ${\displaystyle f}$ is said to be continuous at the point ${\displaystyle x_{0}}$ when the following holds: For any positive real number ${\displaystyle \varepsilon >0,}$ however small, there exists some positive real number ${\displaystyle \delta >0}$ such that for all ${\displaystyle x}$ in the domain of ${\displaystyle f}$ with ${\displaystyle x_{0}-\delta <x<x_{0}+\delta ,}$ the value of ${\displaystyle f(x)}$ satisfies

$${\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .}$$

Alternatively written, continuity of ${\displaystyle f:D\to \mathbb {R} }$ at ${\displaystyle x_{0}\in D}$ means that for every ${\displaystyle \varepsilon >0,}$ there exists a ${\displaystyle \delta >0}$ such that for all ${\displaystyle x\in D}$:

$${\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .}$$

More intuitively, we can say that if we want to get all the ${\displaystyle f(x)}$ values to stay in some small neighborhood around ${\displaystyle f\left(x_{0}\right),}$ we need to choose a small enough neighborhood for the ${\displaystyle x}$ values around ${\displaystyle x_{0}.}$ If we can do that no matter how small the ${\displaystyle f(x_{0})}$ neighborhood is, then ${\displaystyle f}$ is continuous at ${\displaystyle x_{0}.}$

In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

Weierstrass had required that the interval ${\displaystyle x_{0}-\delta <x<x_{0}+\delta }$ be entirely within the domain ${\displaystyle D}$, but Jordan removed that restriction.

Definition in terms of control of the remainder

In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function ${\displaystyle C:[0,\infty )\to [0,\infty ]}$ is called a control function if

• C is non-decreasing
• ${\displaystyle \inf _{\delta >0}C(\delta )=0}$

A function ${\displaystyle f:D\to R}$ is C-continuous at ${\displaystyle x_{0}}$ if there exists such a neighbourhood ${\textstyle N(x_{0})}$ that

$${\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})}$$

A function is continuous in ${\displaystyle x_{0}}$ if it is C-continuous for some control function C.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions ${\displaystyle {\mathcal {C}}}$ a function is ${\displaystyle {\mathcal {C}}}$-continuous if it is ${\displaystyle C}$-continuous for some ${\displaystyle C\in {\mathcal {C}}.}$ For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions

$${\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}}$$

respectively

$${\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}.}$$

Definition using oscillation

The failure of a function to be continuous at a point is quantified by its oscillation.

Continuity can also be defined in terms of oscillation: a function f is continuous at a point ${\displaystyle x_{0}}$ if and only if its oscillation at that point is zero;^[10] in symbols, ${\displaystyle \omega _{f}(x_{0})=0.}$ A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ${\displaystyle \varepsilon }$ (hence a ${\displaystyle G_{\delta }}$ set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.^[11]

The oscillation is equivalent to the ${\displaystyle \varepsilon -\delta }$ definition by a simple re-arrangement and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ${\displaystyle \varepsilon _{0}}$ there is no ${\displaystyle \delta }$ that satisfies the ${\displaystyle \varepsilon -\delta }$ definition, then the oscillation is at least ${\displaystyle \varepsilon _{0},}$ and conversely if for every ${\displaystyle \varepsilon }$ there is a desired ${\displaystyle \delta ,}$ the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

Definition using the hyperreals

Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.

A real-valued function f is continuous at x if its natural extension to the hyperreals has the property that for all infinitesimal dx, ${\displaystyle f(x+dx)-f(x)}$ is infinitesimal^[12]

(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

A useful lemma

Let ${\displaystyle f(x)}$ be a function that is continuous at a point ${\displaystyle x_{0},}$ and ${\displaystyle y_{0}}$ be a value such ${\displaystyle f\left(x_{0}\right)\neq y_{0}.}$ Then ${\displaystyle f(x)\neq y_{0}}$ throughout some neighbourhood of ${\displaystyle x_{0}.}$^[13]

Proof: By the definition of continuity, take ${\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0}$ , then there exists ${\displaystyle \delta >0}$ such that

$${\displaystyle \left|f(x)-f(x_{0})\right|<{\frac {\left|y_{0}-f(x_{0})\right|}{2}}\quad {\text{ whenever }}\quad |x-x_{0}|<\delta }$$

Suppose there is a point in the neighbourhood ${\displaystyle |x-x_{0}|<\delta }$ for which ${\displaystyle f(x)=y_{0};}$ then we have the contradiction

$${\displaystyle \left|f(x_{0})-y_{0}\right|<{\frac {\left|f(x_{0})-y_{0}\right|}{2}}.}$$

Intermediate value theorem

The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:

If the real-valued function f is continuous on the closed interval ${\displaystyle [a,b],}$ and k is some number between ${\displaystyle f(a)}$ and ${\displaystyle f(b),}$ then there is some number ${\displaystyle c\in [a,b],}$ such that ${\displaystyle f(c)=k.}$

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, if f is continuous on ${\displaystyle [a,b]}$ and ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$ differ in sign, then, at some point ${\displaystyle c\in [a,b],}$ ${\displaystyle f(c)}$ must equal zero.

Extreme value theorem

The extreme value theorem states that if a function f is defined on a closed interval ${\displaystyle [a,b]}$ (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists ${\displaystyle c\in [a,b]}$ with ${\displaystyle f(c)\geq f(x)}$ for all ${\displaystyle x\in [a,b].}$ The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval ${\displaystyle (a,b)}$ (or any set that is not both closed and bounded), as, for example, the continuous function ${\displaystyle f(x)={\frac {1}{x}},}$ defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Relation to differentiability and integrability

Every differentiable function

$${\displaystyle f:(a,b)\to \mathbb {R} }$$

is continuous, as can be shown. The converse does not hold: for example, the absolute value function

${\displaystyle f(x)=|x|={\begin{cases}\;\;\ x&{\text{ if }}x\geq 0\\-x&{\text{ if }}x<0\end{cases}}}$

is everywhere continuous. However, it is not differentiable at ${\displaystyle x=0}$ (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.

The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted ${\displaystyle C^{1}((a,b)).}$ More generally, the set of functions

$${\displaystyle f:\Omega \to \mathbb {R} }$$

(from an open interval (or open subset of ${\displaystyle \mathbb {R} }$) ${\displaystyle \Omega }$ to the reals) such that f is ${\displaystyle n}$ times differentiable and such that the ${\displaystyle n}$-th derivative of f is continuous is denoted ${\displaystyle C^{n}(\Omega ).}$ See differentiability class. In the field of computer graphics, properties related (but not identical) to ${\displaystyle C^{0},C^{1},C^{2}}$ are sometimes called ${\displaystyle G^{0}}$ (continuity of position), ${\displaystyle G^{1}}$ (continuity of tangency), and ${\displaystyle G^{2}}$ (continuity of curvature); see Smoothness of curves and surfaces.

Every continuous function

$${\displaystyle f:[a,b]\to \mathbb {R} }$$

is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.

Pointwise and uniform limits

A sequence of continuous functions ${\displaystyle f_{n}(x)}$ whose (pointwise) limit function ${\displaystyle f(x)}$ is discontinuous. The convergence is not uniform.

Given a sequence

$${\displaystyle f_{1},f_{2},\dotsc :I\to \mathbb {R} }$$

of functions such that the limit

$${\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)}$$

exists for all ${\displaystyle x\in D,}$, the resulting function ${\displaystyle f(x)}$ is referred to as the pointwise limit of the sequence of functions ${\displaystyle \left(f_{n}\right)_{n\in N}.}$ The pointwise limit function need not be continuous, even if all functions ${\displaystyle f_{n}}$ are continuous, as the animation at the right shows. However, f is continuous if all functions ${\displaystyle f_{n}}$ are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.

Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. This is often accomplished by specifying when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function ${\displaystyle f:X\to Y}$ is sequentially continuous if whenever a sequence ${\displaystyle \left(x_{n}\right)}$ in ${\displaystyle X}$ converges to a limit ${\displaystyle x,}$ the sequence ${\displaystyle \left(f\left(x_{n}\right)\right)}$ converges to ${\displaystyle f(x).}$ Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If ${\displaystyle X}$ is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ${\displaystyle X}$ is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:^[17]

Proof

Proof. Assume that ${\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }$ is continuous at ${\displaystyle x_{0}}$ (in the sense of ${\displaystyle \epsilon -\delta }$ continuity). Let ${\displaystyle \left(x_{n}\right)_{n\geq 1}}$ be a sequence converging at ${\displaystyle x_{0}}$ (such a sequence always exists, for example, ${\displaystyle x_{n}=x,{\text{ for all }}n}$); since ${\displaystyle f}$ is continuous at ${\displaystyle x_{0}}$ $${\displaystyle \forall \epsilon >0\,\exists \delta _{\epsilon }>0:0<|x-x_{0}|<\delta _{\epsilon }\implies |f(x)-f(x_{0})|<\epsilon .\quad (*)}$$ For any such ${\displaystyle \delta _{\epsilon }}$ we can find a natural number ${\displaystyle \nu _{\epsilon }>0}$ such that for all ${\displaystyle n>\nu _{\epsilon },}$ $${\displaystyle |x_{n}-x_{0}|<\delta _{\epsilon },}$$ since ${\displaystyle \left(x_{n}\right)}$ converges at ${\displaystyle x_{0}}$; combining this with ${\displaystyle (*)}$ we obtain $${\displaystyle \forall \epsilon >0\,\exists \nu _{\epsilon }>0:\forall n>\nu _{\epsilon }\quad |f(x_{n})-f(x_{0})|<\epsilon .}$$ Assume on the contrary that ${\displaystyle f}$ is sequentially continuous and proceed by contradiction: suppose ${\displaystyle f}$ is not continuous at ${\displaystyle x_{0}}$ $${\displaystyle \exists \epsilon >0:\forall \delta _{\epsilon }>0,\,\exists x_{\delta _{\epsilon }}:0<|x_{\delta _{\epsilon }}-x_{0}|<\delta _{\epsilon }\implies |f(x_{\delta _{\epsilon }})-f(x_{0})|>\epsilon }$$ then we can take ${\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0}$ and call the corresponding point ${\displaystyle x_{\delta _{\epsilon }}=:x_{n}}$: in this way we have defined a sequence ${\displaystyle (x_{n})_{n\geq 1}}$ such that $${\displaystyle \forall n>0\quad |x_{n}-x_{0}|<{\frac {1}{n}},\quad |f(x_{n})-f(x_{0})|>\epsilon }$$ by construction ${\displaystyle x_{n}\to x_{0}}$ but ${\displaystyle f(x_{n})\not \to f(x_{0})}$, which contradicts the hypothesis of sequentially continuity. ${\displaystyle \blacksquare }$

Closure operator and interior operator definitions

In terms of the interior operator, a function ${\displaystyle f:X\to Y}$ between topological spaces is continuous if and only if for every subset ${\displaystyle B\subseteq Y,}$

$${\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)~\subseteq ~\operatorname {int} _{X}\left(f^{-1}(B)\right).}$$

In terms of the closure operator, ${\displaystyle f:X\to Y}$ is continuous if and only if for every subset ${\displaystyle A\subseteq X,}$

$${\displaystyle f\left(\operatorname {cl} _{X}A\right)~\subseteq ~\operatorname {cl} _{Y}(f(A)).}$$

That is to say, given any element ${\displaystyle x\in X}$ that belongs to the closure of a subset ${\displaystyle A\subseteq X,}$ ${\displaystyle f(x)}$ necessarily belongs to the closure of ${\displaystyle f(A)}$ in ${\displaystyle Y.}$ If we declare that a point ${\displaystyle x}$ is close to a subset ${\displaystyle A\subseteq X}$ if ${\displaystyle x\in \operatorname {cl} _{X}A,}$ then this terminology allows for a plain English description of continuity: ${\displaystyle f}$ is continuous if and only if for every subset ${\displaystyle A\subseteq X,}$ ${\displaystyle f}$ maps points that are close to ${\displaystyle A}$ to points that are close to ${\displaystyle f(A).}$ Similarly, ${\displaystyle f}$ is continuous at a fixed given point ${\displaystyle x\in X}$ if and only if whenever ${\displaystyle x}$ is close to a subset ${\displaystyle A\subseteq X,}$ then ${\displaystyle f(x)}$ is close to ${\displaystyle f(A).}$

Instead of specifying topological spaces by their open subsets, any topology on ${\displaystyle X}$ can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset ${\displaystyle A}$ of a topological space ${\displaystyle X}$ to its topological closure ${\displaystyle \operatorname {cl} _{X}A}$ satisfies the Kuratowski closure axioms. Conversely, for any closure operator ${\displaystyle A\mapsto \operatorname {cl} A}$ there exists a unique topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (specifically, ${\displaystyle \tau$ :=\{X\setminus \operatorname {cl} A:A\subseteq X\}} ) such that for every subset ${\displaystyle A\subseteq X,}$ ${\displaystyle \operatorname {cl} A}$ is equal to the topological closure ${\displaystyle \operatorname {cl} _{(X,\tau )}A}$ of ${\displaystyle A}$ in ${\displaystyle (X,\tau ).}$ If the sets ${\displaystyle X}$ and ${\displaystyle Y}$ are each associated with closure operators (both denoted by ${\displaystyle \operatorname {cl} }$) then a map ${\displaystyle f:X\to Y}$ is continuous if and only if ${\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))}$ for every subset ${\displaystyle A\subseteq X.}$

Similarly, the map that sends a subset ${\displaystyle A}$ of ${\displaystyle X}$ to its topological interior ${\displaystyle \operatorname {int} _{X}A}$ defines an interior operator. Conversely, any interior operator ${\displaystyle A\mapsto \operatorname {int} A}$ induces a unique topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (specifically, ${\displaystyle \tau$ :=\{\operatorname {int} A:A\subseteq X\}} ) such that for every ${\displaystyle A\subseteq X,}$ ${\displaystyle \operatorname {int} A}$ is equal to the topological interior ${\displaystyle \operatorname {int} _{(X,\tau )}A}$ of ${\displaystyle A}$ in ${\displaystyle (X,\tau ).}$ If the sets ${\displaystyle X}$ and ${\displaystyle Y}$ are each associated with interior operators (both denoted by ${\displaystyle \operatorname {int} }$) then a map ${\displaystyle f:X\to Y}$ is continuous if and only if ${\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)}$ for every subset ${\displaystyle B\subseteq Y.}$^[18]

Filters and prefilters

Continuity can also be characterized in terms of filters. A function ${\displaystyle f:X\to Y}$ is continuous if and only if whenever a filter ${\displaystyle {\mathcal {B}}}$ on ${\displaystyle X}$ converges in ${\displaystyle X}$ to a point ${\displaystyle x\in X,}$ then the prefilter ${\displaystyle f({\mathcal {B}})}$ converges in ${\displaystyle Y}$ to ${\displaystyle f(x).}$ This characterization remains true if the word "filter" is replaced by "prefilter."^[16]