Continuity at a point

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , let $E\subseteq X$, and let $f:E\to Y$. The function $f$ is continuous at a point $x_0\in E$ if

Equivalently, $f$ is continuous at $x_0$ iff $\lim_{x\to x_0} f(x)=f(x_0)$ (see limit of a function ).

Continuity means that small changes in input near $x_0$ produce small changes in output. It is the basic regularity notion in analysis.

Examples:

• $f(x)=x^2$ is continuous at every $x_0\in\mathbb{R}$.
• The indicator function $\mathbf{1}_{\mathbb{Q}}:\mathbb{R}\to\mathbb{R}$ is discontinuous at every point.
• The step function $\mathbf{1}_{[0,\infty)}$ is discontinuous at $0$ but continuous at every $x_0\ne 0$.