# 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.
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Title | global versus local continuity |
---|---|

Canonical name | GlobalVersusLocalContinuity |

Date of creation | 2013-03-22 19:09:07 |

Last modified on | 2013-03-22 19:09:07 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 4 |

Author | CWoo (3771) |

Entry type | Result |

Classification | msc 54C05 |

Classification | msc 26A15 |