Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. Let $\mathcal L$ be the set of linear functions $\mathbb R \to \mathbb R$.

Define the roughness $\mathcal Rf(x)$ of $f$ at $x \in \mathbb R^n$ by

$$\inf_{L \in \mathcal L} \limsup_{y \to x} \left | \frac{f(y) - f(x) - L(y-x)}{|y - x|} \right |.$$

We say that $f$ is *pseudo differentiable* if $\mathcal Rf(x) < \infty$ for all $x \in \mathbb R^n$.

In other words, $f$ is pseudo differentiable if the difference quotients approximate the function to within $O(|y - x|)$ everywhere.

**Question:** Is it true that if $f$ is pseudo differentiable, then $f$ is in the Sobolev space $W^{1, 1}_\text{loc}?$

*Remarks:*

Note that $f$ is differentiable at $x$ if and only if $\mathcal Rf(x)$ is $0$.

In one dimension, the condition that $f$ is pseudo differentiable is equivalent to the upper and lower Dini derivatives being finite everywhere. In this case I believe pseudo differentiable implies (locally) absolutely continuous, and hence $W^{1,1}_\text{loc}$.

In the definition of pseudo differentiable, the condition $Rf(x) < \infty$ holds for all $x$! (Instead of merely almost all $x$)