Thursday, 30 January 2014

real analysis - Find a differentiable f such that f is not continuous.




I'm trying to solve this problem:




Find a differentiable function f:RR such that f:RR is not continuous at any point of R.




Any hints would be appreciated.



Answer



You are looking for a derivative that is discontinuous everywhere on R. Such a function doesn't exist. Since f is the pointwise limit of continuous functions, it is a Baire class 1 function. A theorem of Baire says that the set of discontinuities of f is a meager subset of R.


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