# Continuity

(Redirected from Continuous)

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 definitions of continuity

We can develop the definition of continuity from the $\delta-\epsilon$ formalism which is usually taught in first year calculus courses to general topological spaces.

### Function of a real variable

The $\delta-\epsilon$ formalism defines 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 x0 and) for any $\varepsilon>0$ there exist δ > 0 such that $|x-x_0| < \delta \implies |f(x)-f(x_0)| < \varepsilon. \,$

Simply stated, the limit $\lim_{x\to x_0} f(x) = f(x_0).$

This definition of continuity extends directly to functions of a complex variable.

### Function on a metric space

A function f from a metric space (X,d) to another metric space (Y,e) is continuous at a point $x_0 \in X$ if for all $\varepsilon > 0$ there exists δ > 0 such that $d(x,x_0) < \delta \implies e(f(x),f(x_0)) < \varepsilon . \,$

If we let Bd(x,r) denote the open ball of radius r round x in X, and similarly Be(y,r) denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back $f^{\dashv}$ $f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \,$

### Function on a topological space

A function f from a topological space (X,OX) to another topological space (Y,OY), 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 $f(U_x)=\{f(x') \mid x' \in U_x\}$. In a variation of this definition, instead of being open sets, Ux and Uy can be taken to be, respectively, a neighbourhood of x and a neighbourhood of y = f(x).

## Continuous function

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 Ox (respectively, a closed subset of X). Some content on this page may previously have appeared on Citizendium.