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Hope this isn't a duplicate.

Let $f : \Bbb R \to \Bbb R$ be a function. Let $D(f) = \{x \in \Bbb R : f$ is discontinuous at $x\} $ and $N(f) = \{x \in \Bbb R : f $ is not differentiable at $x \}$ .

It is easy to prove that $D(f)$ is an $F_{\sigma}$ set i.e. can be expressed as a (at most) Countable union of closed sets (possibly empty). Now considering $N(f)$ , since by definition, $D(f) \subseteq N(f) $ , it follows that $N(f)$ contains an at most Countable union of closed sets(possibly empty). In case we further assume that $f$ is convex , it can be proved that $N(f)$ is at most countable.

But my question is what can be said about the structure of $N(f)$ when $f$ is considered in full generality i.e. all we are assuming is that $f : \Bbb R \to \Bbb R$ is a function and no other added restriction is posed on $f$ .

Alex Ravsky
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  • You need to study Denjoy-Young-Saks theorem. See https://en.wikipedia.org/wiki/Denjoy%E2%80%93Young%E2%80%93Saks_theorem – Paramanand Singh Jan 05 '18 at 04:45
  • @ParamanandSingh I've studied a version of Denjoy's theorem that tells if a function $f : \Bbb R \to \Bbb R$ has one-sided derivatives on $\Bbb R$ \ $A$ where A is a countable subset of $\Bbb R$, then $f$ has derivatives on $\Bbb R$ with at most countably many exceptions. But that does not address my problem here. In the theorem, you are taking assumptions on one-sided derivatives, but I'm not assuming anything except that $f$ is a $\Bbb R \to \Bbb R$ function ! –  Jan 05 '18 at 04:52

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