There is a definition for difference quotients of Sobolev functions I do not understand. Let $U\subset\Omega\subset\mathbb{R}^n$ be open sets such that the closure of $U$ is compact in $\Omega$ and let $h\in\mathbb{R}$ with $0<|h| < \mbox{dist}(U,\partial\Omega)$. Then one defines a function $\partial^h_i u$ on $U$ by
$\partial_i^h u(x) := \frac{u(x+he_i) - u(x)}{h}$,
(where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$), see e.g. [1].
But how can one even define this? One must be able to evaluate $u$ at $x$ and $x+he_i$, but since $u$ is only an $L^p$-function, point evaluation is not even well defined. One needs higher regularity, I assume - an unique continuous representative of $u$ which can be evaluated. But no mention of such higher regularity is made in [1] (and the other texts on PDEs I'm reading) where such difference quotients are used. So I guess that no such higher regularity is needed after all?
Am I missing something obvious here? Why can one define $\partial_i^h u$ in such a way? Could you please give me a hint why this definition works or recommend me some literature on this?
Thank you in advance!
Reference:
[1] Lawrence C. Evans, "Partial Differential Equations", 5.8.2. There, Evans defines $\partial_i^hu$ for 'locally summable functions' but uses it for $W^{1,p}$ functions.