global versus local continuity

In this entry, we establish a very basic fact about continuity:

Proposition 1.

A function $f\mathrm{:}X\mathrm{\to }Y$ between two topological spaces is continuous iff it is continuous at every point $x\mathrm{\in }X$.

Proof.

Suppose first that $f$ is continuous, and $x\in X$. Let $f(x)\in V$ be an open set in $Y$. We want to find an open set $x\in U$ in $X$ such that $f(U)\subseteq V$. Well, let $U=f^{-1}(V)$. So $U$ is open since $f$ is continuous, and $x\in U$. Furthermore, $f(U)=f(f^{-1}(V))=V$.

On the other hand, if $f$ is not continuous at $x\in X$. Then there is an open set $f(x)\in V$ in $Y$ such that no open sets $x\in U$ in $X$ have the property

$$f(U)\subseteq V.$$(1)

Let $W=f^{-1}(V)$. If $W$ is open, then $W$ has the property $(1)$ above, a contradiction. Since $W$ is not open, $f$ is not continuous.∎