Continuity at a point

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.

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 $A$ .

In metric spaces, $f$ is continuous at $a$ if and only if $x_n\to a$ implies $f(x_n)\to f(a)$ .