Continuous function

In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon, delta definition of a limit was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.

A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.

As a practical example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

A form of the epsilon, delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of \(y = f(x)\) as follows: an infinitely small increment \(\alpha\) of the independent variable \(x\) always produces an infinitely small change \(f(x+\alpha)-f(x)\) of the dependent variable \(y\) (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity).

The formal definitions for, and distinction between, pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work was not published until the 1930s. Like Bolzano, Karl Weierstrass considered that a function \(y = f(x)\) at a point \(x=c\), that is \(f(x)\big|_{x=c}\) , is continuous if and only if the values of \(f(c)\), \(f(x)\big|_{x\to c^+}\), and \(f(x)\big|_{x\to c^-}\) are all defined and equal. Édouard Goursat assumed continuity provided that the function is defined at \(f(c)\) and is equal to at least one side of the limit \(f(x)\big|_{x\to c}\) , whereas Camille Jordan allowed it even if the function was defined only at \(x=c\). All three of these nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.

A real function (that is, a function from real numbers to real numbers) can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.

Continuity of real functions is usually defined in terms of limits. A function f with variable x is continuous at the real number c if the limit of \(f(x),\) as x tends to c, is equal to \(f(c).\)

There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.

A function is continuous on an open interval if (1) the interval is contained in the function's domain, and (2) the function is continuous at every point in the interval. A function that is continuous on the interval \((-\infty, +\infty)\) (the whole real line) is often simply called a continuous function; one also says that such a function is continuous everywhere. For example, all polynomial functions are continuous everywhere.

A function is continuous on a semi-open or a closed interval if (1) the interval is contained in the function's domain, (2) the function is continuous at every point in the interval, and (3) the function is continuous at the closed endpoints, that is, technically, the value of the function at a closed endpoint equals the limit of the values of the function as the variable approaches the endpoint from the interior of the interval. For example, the function \(f(x) = \sqrt{x}\) is continuous on its whole domain, which is the semi-open interval \([0,+\infty).\)

Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except for some isolated points. Examples include the reciprocal function \(f(x)=\frac{1}{x}\) and the tangent function \(f(x)=\tan x\). When these partial functions are continuous on their domain, they are (in some contexts) said to be continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, they are said to be discontinuous.

A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions \(f(x)=\frac{1}{x}\) and \(f(x)=\sin(\frac{1}{x})\) are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0. A point where a function is discontinuous is called a discontinuity. At a point where a function is not defined, and therefore discontinuous, the discontinuity is removable if a value of the function can be chosen at that point to make the function continuous. For example, the function ⁠\(f(x)=x\sin\tfrac 1x\)⁠ has a removable discontinuity at zero, since ⁠\(\textstyle \lim_{x \to 0} x\sin \frac 1x =0\)⁠, and the discontinuity at 0 of ⁠\(f(x)=\sin\tfrac 1x\)⁠ is not removable, since ⁠\(\textstyle \lim_{x \to 0} \sin \frac 1x\)⁠ does not exist.

Condensed: the full section is in Wikipedia.

Proving the continuity of a function by a direct application of the definition is generally not an easy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:

• Every constant function is continuous
• The identity function ⁠\(f(x) = x\)⁠ is continuous
• Addition and multiplication: if the functions ⁠\(f\)⁠ and ⁠\(g\)⁠ are continuous on their respective domains ⁠\(D_f\)⁠ and ⁠\(D_g\)⁠, then their sum ⁠\(f+g\)⁠ and their product ⁠\(f\cdot g\)⁠ are continuous on the intersection ⁠\(D_f\cap D_g\)⁠, where ⁠\(f+g\)⁠ and ⁠\(f\cdot g\)⁠ are defined by ⁠\((f+g)(x)=f(x)+g(x)\)⁠ and ⁠\((f\cdot g)(x)=f(x)\cdot g(x)\)⁠.
• Reciprocal: If the function ⁠\(f\)⁠ is continuous on the domain ⁠\(D_f\)⁠, then its reciprocal ⁠\(\tfrac 1 f\)⁠, defined by ⁠\((\tfrac 1 f)(x)= \tfrac 1{f(x)}\)⁠ is continuous on the domain ⁠\(D_f\setminus f^{-1}(0)\)⁠, that is, the domain ⁠\(D_f\)⁠ from which the points ⁠\(x\)⁠ such that ⁠\(f(x)=0\)⁠ are removed.
• Function composition: If the functions ⁠\(f\)⁠ and ⁠\(g\)⁠ are continuous on their respective domains ⁠\(D_f\)⁠ and ⁠\(D_g\)⁠, then the composition ⁠\(g\circ f\)⁠ defined by ⁠\({{{1}}}\)⁠ is continuous on ⁠\(D_f\cap f^{-1}(D_g)\)⁠, that is, the part of ⁠\(D_f\)⁠ that is mapped by ⁠\(f\)⁠ inside ⁠\(D_g\)⁠.
• The sine and cosine functions (⁠\(\sin x\)⁠ and ⁠\(\cos x\)⁠) are continuous everywhere.
• The exponential function ⁠\(e^x\)⁠ is continuous everywhere.
• The natural logarithm ⁠\(\ln x\)⁠ is continuous on the domain formed by all positive real numbers ⁠\(\{x\mid x>0\}\)⁠.

These rules imply that every polynomial function is continuous everywhere and that a rational function is continuous everywhere where it is defined, if the numerator and the denominator have no common zeros. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

An example of a function for which the above rules are not sufficient is the sinc function, which is defined by ⁠\(\operatorname{sinc}(0)=1\)⁠ and ⁠\(\operatorname{sinc}(x)=\tfrac{\sin x}{x}\)⁠ for ⁠\(x\neq 0\)⁠. The above rules show immediately that the function is continuous for all ⁠\(x\neq 0\)⁠, but to prove continuity at ⁠\(x=0\)⁠, one has to prove \[\lim_{x\to 0} \frac{\sin x}{x} = 1.\] This is indeed true, and thus the sinc function is continuous function on all real numbers.

An example of a discontinuous function is the Heaviside step function \(H\), defined by \[H(x) = \begin{cases} 1 & \text{ if } x \ge 0\\ 0 & \text{ if } x < 0 \end{cases}\]

Pick for instance \(\varepsilon = 1/2\). Then there is no \(\delta\)-neighborhood around \(x = 0\), i.e. no open interval \((-\delta,\;\delta)\) with \(\delta > 0,\) that will force all the \(H(x)\) values to be within the \(\varepsilon\)-neighborhood of \(H(0)\), i.e. within \((1/2,\;3/2)\). Intuitively, we can think of this type of discontinuity as a sudden jump in function values.

Similarly, the signum or sign function \[\sgn(x) = \begin{cases} \;\;\ 1 & \text{ if }x > 0\\ \;\;\ 0 & \text{ if }x = 0\\ -1 & \text{ if }x < 0 \end{cases}\] is discontinuous at \(x = 0\) but continuous everywhere else. Yet another example: the function \[f(x) = \begin{cases} \sin\left(x^{-2}\right)&\text{ if }x \neq 0\\ 0&\text{ if }x = 0 \end{cases}\] is continuous everywhere except \(x = 0\).

Besides plausible continuities and discontinuities like above, there are also functions with pathological behavior; for example, Thomae's function, \[f(x)=\begin{cases} 1 &\text{ if } x=0\\ \frac{1}{q}&\text{ if } x = \frac{p}{q} \text{(in lowest terms) is a rational number}\\ 0&\text{ if }x\text{ is irrational}. \end{cases}\] is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers, \[D(x)=\begin{cases} 0&\text{ if }x\text{ is irrational } (\in \R \setminus \Q)\\ 1&\text{ if }x\text{ is rational } (\in \Q) \end{cases}\] is nowhere continuous.

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right- and left-continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, \(f\) is said to be right-continuous at the point \(c\) if the following holds: For any number \(\varepsilon > 0\) however small, there exists some number \(\delta > 0\) such that for all x in the domain with \(c < x < c + \delta,\) the value of \(f(x)\) satisfies \[|f(x) - f(c)| < \varepsilon\]

This is the same condition as continuous functions, except it is required to hold only for \(x\) strictly larger than \(c\). Requiring \(|f(x) - f(c)| < \varepsilon\) to hold instead for all \(x\) with \(c - \delta < x < c\) yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.

A function \(f\) is lower semi-continuous at the point \(c\) if, roughly, any jumps that might occur only go down, but not up. That is, for any \(\varepsilon > 0,\) there exists some number \(\delta > 0\) such that for all x in the domain with \(|x - c| < \delta,\) the value of \(f(x)\) satisfies \[f(x) \geq f(c) - \varepsilon.\] The reverse condition is upper semi-continuity.

The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set \(X\) equipped with a function (called a metric) \(d_X,\) that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function \[d_X : X \times X \to \R\] that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces \(\left(X, d_X\right)\) and \(\left(Y, d_Y\right)\) and a function \[f : X \to Y\] then \(f\) is continuous at the point \(c \in X\) (with respect to the given metrics) if for any positive real number \(\varepsilon > 0,\) there exists a positive real number \(\delta > 0\) such that all \(x \in X\) satisfying \(d_X(x, c) < \delta\) will also satisfy \(d_Y(f(x), f(c)) < \varepsilon.\) As in the case of real functions above, this is equivalent to the condition that for every sequence \(\left(x_n\right)\) in \(X\) with \(\lim x_n = c,\) we have \(\lim f\left(x_n\right) = f(c).\) The latter condition can be weakened as follows: \(f\) is continuous at the point \(c\) if and only if for every convergent sequence \(\left(x_n\right)\) in \(X\) with limit \(c\), the sequence \(\left(f\left(x_n\right)\right)\) is a Cauchy sequence, and \(c\) is in the domain of \(f\).

The set of points at which a function between metric spaces is continuous is a \(G_{\delta}\) set, this follows from the \(\varepsilon-\delta\) definition of continuity.

This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator \[T : V \to W\] between normed vector spaces \(V\) and \(W\) (which are vector spaces equipped with a compatible norm, denoted \(\|x\|\)) is continuous if and only if it is bounded, that is, there is a constant \(K\) such that \[\|T(x)\| \leq K \|x\|\] for all \(x \in V.\)

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way \(\delta\) depends on \(\varepsilon\) and \(c\) in the definition above. Intuitively, a function \(f\) as above is uniformly continuous if the \(\delta\) does not depend on the point \(c\). More precisely, it requires that for every real number \(\varepsilon > 0\) there exists \(\delta > 0\) such that for every \(c, b \in X\) with \(d_X(b, c) < \delta,\) the inequality \(d_Y(f(b), f(c)) < \varepsilon\) holds. Thus, any uniformly continuous function is continuous. The converse does not hold generally, but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.

A function is Hölder continuous with exponent \(\alpha\) (a real number) if there is a constant \(K\) such that for all \(b, c \in X,\) the inequality \[d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha\] holds. Any Hölder continuous function is uniformly continuous. The particular case \(\alpha = 1\) is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant \(K\) such that the inequality \[d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)\] holds for any \(b, c \in X.\) The Lipschitz condition occurs, for example, in the Picard-Lindelöf theorem concerning the solutions of ordinary differential equations.

Another, more abstract, notion of continuity is the continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set \(X\) together with a topology on \(X\), which is a set of subsets of \(X\) satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing one to talk about the neighborhoods of a given point. The elements of a topology are called open subsets of \(X\) (with respect to the topology).

A function \[f : X \to Y\] between two topological spaces \(X\) and \(Y\) is continuous if for every open set \(V \subseteq Y,\) the inverse image \[f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}\] is an open subset of \(X\). That is, \(f\) is a function between the sets \(X\) and \(Y\) (not on the elements of the topology \(T_X\)), but the continuity of \(f\) depends on the topologies used on \(X\) and \(Y\).

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in \(Y\) are closed in \(X\).

An extreme example: if a set \(X\) is given the discrete topology (in which every subset is open), all functions \[f : X \to T\] to any topological space \(T\) are continuous. On the other hand, if \(X\) is equipped with the indiscrete topology (in which the only open subsets are the empty set and \(X\)) and the space \(T\) set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

The translation in the language of neighborhoods of the \((\varepsilon, \delta)\)-definition of continuity leads to the following definition of the continuity at a point:

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images. One of those ways is the following. As every set that contains a neighborhood is also a neighborhood, and \(f^{-1}(V)=U\) is the largest subset \(U\subseteq X\) such that \(f(U) \subseteq V,\) the above definition may be simplified into:

As an open set is a set that is a neighborhood of all its points, a function \(f : X \to Y\) is continuous at every point of \(X\) if and only if it is a continuous function.

If \(X\) and \(Y\) are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at \(x\) and \(f(x)\) instead of all neighborhoods. This gives back the above \(\varepsilon-\delta\) definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a Hausdorff space, it is still true that \(f\) is continuous at \(a\) if and only if the limit of \(f\) as \(x\) approaches \(a\) is \(f(a)\). At an isolated point, every function is continuous.

Given \(x \in X,\) a map \(f : X \to Y\) is continuous at \(x\) if and only if whenever \(\mathcal{B}\) is a filter on \(X\) that converges to \(x\) in \(X,\) which is expressed by writing \(\mathcal{B} \to x,\) then necessarily \(f(\mathcal{B}) \to f(x)\) in \(Y\). If \(\mathcal{N}(x)\) denotes the neighborhood filter at \(x\) then \(f : X \to Y\) is continuous at \(x\) if and only if \(f(\mathcal{N}(x)) \to f(x)\) in \(Y\). Moreover, this happens if and only if the prefilter \(f(\mathcal{N}(x))\) is a filter base for the neighborhood filter of \(f(x)\) in \(Y\).

If \(f : X \to Y\) and \(g : Y \to Z\) are continuous, then so is the composition \(g \circ f : X \to Z.\) If \(f : X \to Y\) is continuous and

• \(X\) is compact, then \(f(X)\) is compact.
• \(X\) is connected, then \(f(X)\) is connected.
• \(X\) is path-connected, then \(f(X)\) is path-connected.
• \(X\) is Lindelöf, then \(f(X)\) is Lindelöf.
• \(X\) is separable, then \(f(X)\) is separable.

The possible topologies on a fixed set \(X\) are partially ordered: a topology \(\tau_1\) is said to be coarser than another topology \(\tau_2\) (notation: \(\tau_1 \subseteq \tau_2\)) if every open subset with respect to \(\tau_1\) is also open with respect to \(\tau_2.\) Then, the identity map \[\operatorname{id}_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)\] is continuous if and only if \(\tau_1 \subseteq \tau_2\) (see also comparison of topologies). More generally, a continuous function \[\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)\] stays continuous if the topology \(\tau_Y\) is replaced by a coarser topology and/or \(\tau_X\) is replaced by a finer topology.

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. If an open map \(f\) has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function \(f\) between two topological spaces, the inverse function \(f^{-1}\) need not be continuous. A bijective continuous function with a continuous inverse function is called a homeomorphism.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Given a function \[f : X \to S,\] where \(X\) is a topological space and \(S\) is a set (without a specified topology), the final topology on \(S\) is defined by letting the open sets of \(S\) be those subsets \(A \subset S\) for which \(f^{-1}(A)\) is open in \(X\). If \(S\) has an existing topology, \(f\) is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on \(S\). Thus, the final topology is the finest topology on \(S\) that makes \(f\) continuous. If \(f\) is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by \(f\).

Dually, for a function \(f\) from a set \(S\) to a topological space \(X\), the initial topology on \(S\) is defined by designating as an open set every subset \(A \subset S\) such that \(A = f^{-1}(U)\) for some open subset \(U\) of \(X\). If \(S\) has an existing topology, \(f\) is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on \(S\). Thus, the initial topology is the coarsest topology on \(S\) that makes \(f\) continuous. If \(f\) is injective, this topology is canonically identified with the subspace topology of \(S\), viewed as a subset of \(X\).

A topology on a set S is uniquely determined by the class of all continuous functions \(S \to X\) into all topological spaces \(X\). Dually, a similar idea can be applied to maps \(X \to S.\)