Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. 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 to sufficiently small changes of its argument. 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 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 an 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account at time $t$ would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

History[edit]

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 definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,^[1] Karl Weierstrass^[2] denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat^[3] allowed the function to be defined only at and on one side of c, and Camille Jordan^[4] allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.^[5] 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.^[6]

$D=\mathbb {R}$ : i.e., $D$ is the whole set of real numbers. or, for $a$ and $b$ real numbers,

$D=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\}$ : $D$ is a , or

closed interval

$D=(a,b)=\{x\in \mathbb {R} \mid a<x<b\}$ : $D$ is an .

open interval

X is , then f(X) is compact.

compact

X is , then f(X) is connected.

connected

X is , then f(X) is path-connected.

path-connected

X is , then f(X) is Lindelöf.

Lindelöf

X is , then f(X) is separable.

separable

Related notions[edit]

If $f:S\to Y$ is a continuous function from some subset $S$ of a topological space $X$ then a continuous extension of $f$ to $X$ is any continuous function $F:X\to Y$ such that $F(s)=f(s)$ for every $s\in S,$ which is a condition that often written as $f=F{\big \vert }_{S}.$ In words, it is any continuous function $F:X\to Y$ that restricts to $f$ on $S.$ This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If $f:S\to Y$ is not continuous, then it could not possibly have a continuous extension. If $Y$ is a Hausdorff space and $S$ is a dense subset of $X$ then a continuous extension of $f:S\to Y$ to $X,$ if one exists, will be unique. The Blumberg theorem states that if $f:\mathbb {R} \to \mathbb {R}$ is an arbitrary function then there exists a dense subset $D$ of $\mathbb {R}$ such that the restriction $f{\big \vert }_{D}:D\to \mathbb {R}$ is continuous; in other words, every function $\mathbb {R} \to \mathbb {R}$ can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in order theory, an order-preserving function $f:X\to Y$ between particular types of partially ordered sets $X$ and $Y$ is continuous if for each directed subset $A$ of $X,$ we have $\sup f(A)=f(\sup A).$ Here $\,\sup \,$ is the supremum with respect to the orderings in $X$ and $Y,$ respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.^[19]^[20]

In category theory, a functor $F:{\mathcal {C}}\to {\mathcal {D}}$ between two categories is called continuous if it commutes with small limits. That is to say, $\varprojlim _{i\in I}F(C_{i})\cong F\left(\varprojlim _{i\in I}C_{i}\right)$ for any small (that is, indexed by a set $I,$ as opposed to a class) diagram of objects in ${\mathcal {C}}$ .

A continuity space is a generalization of metric spaces and posets,^[21]^[22] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.^[23]