On approximate differentiability of the maximal function

We prove that if $f\in L^1(\mathbb{R}^n)$ is approximately differentiable a.e., then the Hardy-Littlewood maximal function $\mathcal{M}f$ is also approximately differentiable a.e. Moreover, if we only assume that $f\in L^1(\mathbb{R}^n)$, then any open set of $\mathbb{R}^n$ contains a subset of positive measure such that $\mathcal{M} f$ is approximately differentiable on that set. On the other hand we present an example of $f\in L^1(\mathbb{R})$ such that $\mathcal{M}f$ is not approximately differentiable a.e.