Continuity at a point - Imported Knowlpedia Content

Let $f:(X,d_X)\to(Y,d_Y)$ and let $a\in X$ . We say $f$ is continuous at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for all $x\in X$ ,

d_X(x,a)<\delta \quad\Rightarrow\quad d_Y\!\bigl(f(x),f(a)\bigr)<\varepsilon.

In words: points $x$ sufficiently close to $a$ must have images $f(x)$ close to $f(a)$ .

Examples

Examples:

$f(x)=x^2$ is continuous at every $a\in\mathbb{R}$ .
The step function $f(x)=\mathbf{1}_{(0,\infty)}(x)$ is not continuous at $0$ .

Remarks

If $f$ is continuous at every point of a set $A$ , then $f$ is continuous on $A$ .

Sequential characterization (metric spaces): $f$ is continuous at $a$ iff whenever $x_n\to a$ , we have $f(x_n)\to f(a)$ . This reframes continuity using limits of sequences.