I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words:
Let $U \subset \mathbb{R}$ open. Is there a differentiable function $f:U \to \mathbb{R}$ where $\{x \in U :f'(x) = 0 \}$ has empty interior and is of second category?
It seems the following.
We shall use the construction of Pompeiu derivative, that is a real-valued function of one real variable that is the derivative of an everywhere differentiable function $f$ and that vanishes in a dense set $Z=\{x\in [0,1]:f′(x)=0\}$. Then $Z$ is a dense $G_\delta$ subset of the segment $[0,1]$, that is of second category.
Now we have to to assure that the set $Z$ has an empty interior. Let $\{x_i\}_{i\in\Bbb N}$ be an enumeration of a countable set $X$ of the irrational numbers dense in the unit segment $[0,1]$. In Pompeiu's construction of the function $g$ put $$a_j=2^{-j}\min_{i\le j}\{(x_i-q_j)^{2/3}\}.$$ Then $g’(x_i)$ is finite for each index $i$. So the function $f=g^{-1}$ has a finite non-zero derivative at a point $g(x_i)$ for each index $i$. Since the set $X$ is dense in the unit segment $[0,1]$ and the function $g$ is a homeomorphism, the set $g(X)$ is dense in $[0,1]$. Hence $$\text{int} Z\subset \text{int} ([0,1]\setminus g(X))=\varnothing.$$