I'm trying to solve this problem:
Find a differentiable function f:R⟶R such that f′:R⟶R 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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