I was doing some fairly simple research a few hours ago and I almost asked a similar question with the word continuous instead of differentiable in the title, but then I found this question asked by Gro-Tsen where there is an affirmative answer to that question.
Apparently, that is the result of Blumberg, that for every $f: \mathbb R \to \mathbb R$ there exists a dense subset $D$ of $\mathbb R$ such that $f|_D$ is continuous.
Blumberg´s paper can be found here and I have slightly did a research of his arguments, however, I am not sure can they be adapted to show that $f$ is differentiable at at least one point when restricted to some everywhere dense subset of $\mathbb R$.
Honestly, I expect that there are some $f$´s which have the property that when restricted to every possible everywhere dense subset of $\mathbb R$ are non-differentiable everywhere on all such sets
However, I am not sure, and that´s why I ask it here, since I think that´s known, because Blumberg´s result is relatively long time ago established (1922).
Here is the question:
- Is it true that for every function $f: \mathbb R \to \mathbb R$ there exists at least one everywhere dense set $D \subseteq \mathbb R$ such that $f|_D$ is differentiable at at least one point?