constant functions and continuity

It is easy to see that every constant function between topological spacesMathworldPlanetmath is continuousMathworldPlanetmathPlanetmath. A converseMathworldPlanetmath result is as follows.


Suppose X is path connected and D is a countableMathworldPlanetmath discrete topological space. If f:XD is continuous, then f is a constant function.


By this result ( we can assume that D is either {1,,n}, n2 or , and these are equipped with the subspace topology of . Suppose f(X) has at least two distinct elements, say α,β so that


for some x,yX. Since X is path connected there is a continuous path γ:[0,1]X such that γ(0)=x and γ(1)=y. Then fγ:[0,1]D is continuous. Since D has the subspace topology of , this result ( implies that also fγ:[0,1] is continuous. Since fγ achieves two different values, it achieves uncountably many values, by the intermediate value theorem. This is a contradictionMathworldPlanetmathPlanetmath since fγ([0,1]) is countable. ∎

Title constant functions and continuity
Canonical name ConstantFunctionsAndContinuity
Date of creation 2013-03-22 15:17:31
Last modified on 2013-03-22 15:17:31
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Theorem
Classification msc 03E20