global versus local continuity

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

Proposition 1.

A function f:XY between two topological spacesMathworldPlanetmath is continuousMathworldPlanetmathPlanetmath iff it is continuous at every point xX.


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

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

f(U)V. (1)

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

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