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?

continuousfunction is nowhere differentiable, then so should be its restriction to any dense set. The idea is that if the slopes of secant lines behave badly near $x$, then continuity forces them to behave just as badly on a dense set near $x$. In particular, @LiviuNicolaescu's suggestion should work. $\endgroup$ – Will Brian Apr 2 '20 at 12:31Restriction theorems in real analysis(preprint version here). $\endgroup$ – Dave L Renfro Apr 2 '20 at 17:47