Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that

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

I would like to ask for a reference of the following statement, i.e.,

If $f$ is locally Lipschitz, then the Hausdorff dimension of $E$ is at most $n-1$.

My closest search is the following

Definition 1.2. A set $E \subset \mathbb{R}^n$ is porous at a point $x \in E$ if there is a $c>0$ and there is a sequence $y_n \rightarrow 0$ such that the balls $B\left(x+y_n, c\left|y_n\right|\right)$ are disjoint from $E$. The set $E$ is porous if it is porous at each of its points, and it is called $\sigma$-porous if it is a countable union of porous sets.

Theorem 1.3. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a Lipschitz function. Then the set of those points at which $f$ is not differentiable but it is (directionally) differentiable in $n$ linearly independent directions is $\sigma$-porous.

We have the Hausdorff dimension of a $\sigma$-porous subset of $\mathbb R^n$ is at most $n-1$. However, the Lipschitz function in Theorem 1.3. has one more restriction, i.e., it must be directionally differentiable in $n$ linearly independent directions.

Could you elaborate on such a reference?