Wednesday, 24 April 2013

real analysis - Give an example of a function uniformly differentiable with unbounded derivative




Give an example of a function f:UR

uniformly differentiable with unbounded derivative f, UR open set.



Any hints would be appreciated.


Answer



Take f(x)=x3/2 on U=[0,). Then f(x)=(3/2)x1/2, which is uniformly continuous on U. By the MVT, f is uniformly differentiable on [0,). Since f is unbounded, we have an example.


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