Suppose that is such that, for some , if , then . Prove that is constant.Problem Statement:

I dont see how this is true. for instance, . Is there something I am missing?

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- Nov 7th 2010, 02:10 PMDontKnoMaffanalysis
Suppose that is such that, for some , if , then . Prove that is constant.__Problem Statement:__

I dont see how this is true. for instance, . Is there something I am missing? - Nov 7th 2010, 02:34 PMAlso sprach Zarathustra
I think it is something with the derivative... f'(x)=|f(x)-f(y)|/|x-y|=0 when epsilon is very small.

f'(x)=0 for all x ==> f(x) constant for all x.

Hmmm... - Nov 7th 2010, 05:40 PMDrexel28
You could state this a little bit more rigorously.

1) Let then the above says that . Thus, since we have that (by continuity of ) and thus by the squeeze theorem is differentiable at and . Since was arbitrary it follows that for all

2) Since is an__interval__we may conclude that .

Just a comment though.