Suppose $f,g:[a,b]\rightarrow \mathbb C,\ \ f=g $ $ \ $ a.e. and that $f$ is differentiable a.e. Does it follow that $g$ is a.e. differentiable and $g'=f'$ a.e.? If not, is there any good counterexample? By the way, I am considering the Lebesgue measure here.
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