Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and $$ E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}. $$

Then we have the following result which is

Theorem: If $X= \mathbb R^d$ and $f$ is convex, then the Hausdorff dimension of $E$ is at most $d-1$.

Differentiability is a local property, so I guess above theorem is true even though $X \neq \mathbb R^d$. Can we extend above theorem to obtain below one?

If $f$ is convex, then the Hausdorff dimension of $E$ is at most $d-1$.