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If a function $g$ on $\mathbf{R}$ is everywhere differentiable, why is $f=g'$ the limit of a pointwise convergent sequence of continuous functions $f_n$?

More generally, does this also hold for any function $f$ on $\mathbf{R}$ possessing the intermediate value property?

Aubrey
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1 Answers1

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Try $f_n(x)=n\cdot(g(x+1/n)-g(x))$.

Did
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  • Great, thank you! Do you think it also holds for any function that has the mean value property? I don't think so. – Aubrey Dec 31 '13 at 10:30
  • By the way, what do YOU think? – Did Dec 31 '13 at 10:40
  • Sorry for the typo---I meant "intermediate value property" but I had written "mean" instead of "intermediate." The Wikipedia page on Darboux functions says that any function on the real line is the sum of two Darboux functions. There exist, however, functions which are not of Baire class 1. It follows that not every Darboux function is of Baire class 1, but I cannot prove this. – Aubrey Dec 31 '13 at 14:02
  • The Conway base 13 function is an everywhere discontinuous Darboux function, so it is not of Baire class 1, by the Baire Characterization Theorem. – Aubrey Dec 31 '13 at 14:16