Saturday, 24 January 2015

real analysis - Function that is not uniformly differentiable



Define f:AR (f differentiable)to be uniformly differentiable if and only if for ε>0 there exists δ>0 such that



|h|<δ|f(x+h)f(x)hf(x)|<ε




I am looking for an example of f that is not uniformly differentiable. My idea was to choose f with f sufficiently steep. For example, f(x)=1x. But on [1,)



|f(x+h)f(x)hf(x)|=|hx2(x+h)||hx+h|



and I don't know how to use this. On (0,1) I similarly can't find a lower bound on this either. Maybe 1/x is in fact uniformly differentiable but I coudln't bound it from above either.



My questions are: Is 1/x uniformly differentiable or not and how to show it. Also can you please give me an example of f that is not uniformly differentiable.


Answer



Let f(x)=1/x. As you showed,




|f(x+h)f(x)hf(x)|=|h(x+h)x2|
For any δ>0, you can just let h=x=1M, for some sufficiently large M.Then
|f(x+h)f(x)hf(x)|=|1/M(2/M)(1/M)2|=M22

so this function is certainly not uniformly differentiable.
The point is that the condition for uniform differentiability requires the same value of \boldsymbol\delta for all \boldsymbol{x, h}, i.e. \delta does not depend on x and h.


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