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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:

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

  2. 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 $W^{1,1}_\text{loc}$.

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

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    $\begingroup$ A slightly easier way to state this condition is to say that $f$ is pseudo-differentiable at $x$ if $\limsup |f(y)-f(x)|/|y-x|<\infty$ (or maybe it would be more descriptive to call this locally Lipschitz continuous at $x$). $\endgroup$ Jun 18, 2022 at 15:26

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The function $$ f(x) = \begin{cases} x\sin 1/x^2 & x\not= 0 \\ 0 & x=0 \end{cases} $$ gives a counterexample. We have $f\in C^{\infty}(U)$ when we restrict to $U=\mathbb R\setminus \{ 0\}$, so if $f$ had a distributional derivative in $L^1_{\textrm{loc}}$, it would have to be its classical derivative $f'=-(2/x^2)\cos (1/x^2)+\sin (1/x^2)$, but this fails to be integrable near $x=0$.

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    $\begingroup$ Or $f=x^2\sin (1/x^3)$, which is even differentiable everywhere (but not $C^1$, obviously, or it would be in $W^{1,1}$). $\endgroup$ Jun 18, 2022 at 15:51
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    $\begingroup$ Ah so a function can be differentiable everywhere yet it’s derivative fails to be in $L^1$. Thanks for the example! $\endgroup$
    – Nate River
    Jun 18, 2022 at 15:53

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