Is it true that the definition of [approximate differentiability presented here](https://www.encyclopediaofmath.org/index.php/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 $L:\mathbb{R}^N \to \mathbb{R}$.
> 
> 
> 



Equivalent notion of approximate differentiability
I think the answer is no.


First of all consider a related question: is [the standard definition of approximate continuity](https://www.encyclopediaofmath.org/index.php/Approximate_continuity) of $g\colon \mathbb R \to \mathbb R$ equivalent to the following one


$$
\lim\_{r \to 0} \rlap{-}\!\!\int\_{B\_r(x)} \min \left\{|g(y)-g(x)|,1 \right\} dy = 0.
$$


This definition is in fact strictly weaker. Consider for instance the function $g(x) = \sum\_{n=1}^\infty b\_n \chi\_{[-r\_n, -r\_n + a\_n]}$, where $r\_n = 3^{-n}$, $a\_n = \frac12 4^{-n}$, $b\_n = 2^n$ and $x=0$.


Then for $r=r\_n$


$$
\rlap{-}\!\!\int\_{B\_r(x)} \min \left\{|g(y)-g(x)|,1 \right\} dy = \frac{1}{2 r\_n}\sum\_{k=n}^\infty a\_k = \frac13 \left(\frac34\right)^{n} \to 0
$$


while


$$
\rlap{-}\!\!\int\_{B\_r(x)} |g(y)-g(x)| \, dy = \frac{1}{2 r\_n}\sum\_{k=n}^\infty a\_k b\_k = \frac12 \left(\frac32\right)^{n} \to \infty
$$


as $n\to \infty$.


For the original question one can consider $f(x) = x g(x)$. Let me know if additional details are needed.


Update. For some closely related results see also Proposition 3.65 in *Functions of Bounded Variation* by L. Ambrosio, N. Fusco and D. Pallara (2000).


