Continuous function

In mathematics, a continuous function is, intuitively speaking, a function whose "value" does not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuous function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.

Formal definition

A function f from a topological space $(X,O_{X})$ to another topological space $(Y,O_{Y})$, usually written as $f:(X,O_{X})\rightarrow (Y,O_{Y})$, is a continuous function if for every point $x\in X$ and for every open set $U_{y}\in O_{Y}$ containing the point y=f(x), there exists an open set $U_{x}\in O_{X}$ containing x such that $f(U_{x})\subset U_{y}$. Here $f(U_{x})=\{f(x')\in Y\mid x'\in U_{x}\}$. In a variation of this definition, instead of being open sets, $U_{x}$ and $U_{y}$ can be taken to be, respectively, a neighbourhood of x and a neighbourhood of $y=f(x)$.

An important equivalent definition, but perhaps less convenient to work with directly, is that a function $f:(X,O_{X})\rightarrow (Y,O_{Y})$ is continuous if for any open (respectively, closed) set $U\in O_{Y}$ the set $f^{-1}(U)=\{x\in X\mid f(x)\in U\}$ is an open (respectively, closed) set in $O_{x}$. In this definition, a continuous function is simply a function which maps open sets to open sets or, equivalently, closed sets to closed sets.

The first definition corresponds to a generalization of the $\delta -\epsilon$ argument which are usually taught in first year calculus courses to, among other things, define continuity for functions which map the real numbers to itself.