Is it true that the notion of approximate differentiability of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?

$$\lim_{r \to 0} \rlap{-}\!\!\int_{B_r(x)} \min \left\{\frac{f(y)-f(x) - L(y-x)}{|y-x|},1 \right\} dy = 0$$ for some linear $L:\mathbb{R}^N \to \mathbb{R}$.