Continuity

In mathematics, the notion of continuity of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a 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 of continuity

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 said to be continuous at the point $x\in X$ if 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 $y=f(x)$.

This definition is a generalization of the $\delta -\epsilon$ formalism which are usually taught in first year calculus courses to, among other things, define limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at $x_{0}\in \mathbb {R}$ if (it is defined in a neighborhood of $x_{0}$ and) for any $\varepsilon >0$ there exist $\delta >0$ such that

$|x-x_{0}|\leq \delta \implies |f(x)-f(x_{0})|\leq \varepsilon .$

Simply stated,

$\lim _{x\to x_{0}}f(x)=f(x_{0}).$

Continuous function - global approach

If the function f is continuous at every point $x\in X$ then it is said to be a continuous function. There is another important equivalent definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis. A function $f:(X,O_{X})\rightarrow (Y,O_{Y})$ is said to be continuous if for any open set $U\in O_{Y}$ (respectively, closed subset of Y ) the set $f^{-1}(U)=\{x\in X\mid f(x)\in U\}$ is an open set in $O_{x}$ (respectively, a closed subset of X).