Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function.
Define a *superdifferential* of $f$ at $x\in Q$ by
$$
D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid \text{$f(y)\le f(x)+p\cdot(y-x)+o(|x-y|)$ as $Q\ni y\to x$}\}.
$$

Is a set $$ \Sigma(f)=\{x\in Q \mid D^{+}f(x)=\emptyset\} $$ Lebesgue measurable?

I'm facing the above problem, but am not familiar with Lebesgue measure theory and so I don't know how to verify. Of course, I know that if $f\in C^{1}(Q)$, then $\Sigma(f)=\emptyset$ thus Lebesgue measurable set, and it is clear even if $f\in Lip(Q)$. However, I'm wondering if this is really true for general continuous functions.

This problem may be fundamental, but I'm glad if you teach me how to discuss. In addition, I want to consider the same problem for upper semicontinuous functions and so it is great pleasure if you give some comments.

Thank you in advance.