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Let $f$ there be a real-valued differentiable function everywhere in the interval $]a,b[$.

Does $\frac{df}{dx}$ need to be continuous somewhere in the interval $]a,b[$? Or can a differentiable function $f$ exist so that $\frac{df}{dx}$ is continuous nowhere in the interval $]a,b[$?

NSERC Protester
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1 Answers1

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$$f'(x) = \lim_{n\to\infty} n ( f(x+1/n) - f(x) )$$

This is the limit of continuous functions and thus Baire class $1$, that is, it is the pointwise limit of continuous functions.

A theorem states that a pointwise limit of continuous functions can only have discontinuities at a meagre set and thus must be continuous on a dense set of points.

A.S
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  • You're essentially saying that $f'(x)$ can have discontinuities so long that the discontinuities in $f'(x)$ form a nondense set?

    Does that imply that $f'(x)$ has to be continuous somewhere?

     – NSERC Protester May 08 '15 at 19:02
  • No, I'm saying that the set of points at which $f'(x)$ is discontinuous must be the countable union of nowhere dense sets. This can be a dense set, but its complement must also be dense, so the points of continuity must be dense. – A.S May 08 '15 at 19:04
  • @NSERCProtester To learn more, see http://en.wikipedia.org/wiki/Baire_function – A.S May 08 '15 at 19:06