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# global versus local continuity

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

###### Proposition 1.

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

###### 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. ∎

## Mathematics Subject Classification

54C05*no label found*26A15

*no label found*

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