MathGloss

Let $f: X\to Y$ be a function between topological spaces $X$ and $Y$. Then $f$ is continuous if and only if the preimage of an open set in $Y$ is open in $X$.

A function $f: X\to Y$ between metric spaces $(X,\rho)$ and $(Y,\sigma)$ is continuous at $x\in X$ if and only if for every $\varepsilon > 0$, there exists $\delta> 0$ such that $\sigma(f(x),f(x’)) < \varepsilon$ whenever $\rho(x,x’) < \delta$.

A function $f: X\to Y$ between metric spaces $(X,\rho)$ and $(Y,\sigma)$ is continuous at $x\in X$ if and only if for any sequence ${x_n}{n\in\mathbb N}$in$X$ converging to $x$, ${f(x_n)}{n\in \mathbb N}$converges to$f(x)$.

These three definitions are all equivalent in their respective contexts.

Wikidata ID: Q170058

This site is open source. Improve this page.