Subdifferentiable functions satisfy Lusin properties of class $C^1$ or $C^2$

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a function. Assume that for a measurable set $\mathrm{\Omega }$ and almost every $x\in \mathrm{\Omega }$ there exists a vector ${\xi }_x\in {\mathbb{R}}^n$ such that $$\underset{h\to 0}{\mathrm{lim\hspace{0.17em}inf}}\frac{f(x+h)-f(x)-⟨{\xi }_x,h⟩}{\mathrm{\mid }h{\mathrm{\mid }}^2}>-\mathrm{\infty }\mathrm{.}$$ Then we show that $f$ satisfies a Lusin-type property of order $2$ in $\mathrm{\Omega }$, that is to say, for every $\epsilon >0$ there exists a function $g\in C^2({\mathbb{R}}^n)$ such that ${\mathcal{L}}^n(\{x\in \mathrm{\Omega }:f(x)\mathrm{\ne }g(x)\})\le \epsilon$. In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class $C^2$. We also obtain a similar result (replacing $C^2$ with $C^1$) for the Fr'echet subdifferential. Finally we provide some examples showing that this kind of results are no longer true for “Taylor subexpansions” of higher order.