Is there any continuous function that is only differentiable on $\mathbb{Q}$?

Question

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$.

I already know there is no function that is continuous everywhere in $\mathbb Q$ but nowhere outside of it, but I wonder if something similar can be said about differentiability. If not, how could one prove that?

Thanks in advance!

Answer 1

Here is a reference:

Mark Lynch. A Continuous Function That Is Differentiable Only at the Rationals. Mathematics Magazine 86, no. 2 (2013): 132-35. doi:10.4169/math.mag.86.2.132.

An explicit construction is given of a function that is continuous on an interval, and differentiable only at the rationals.

See also references within (Zahorski's Theorem) for a nonconstructive proof.