In Evans' book on PDE, chapter 5 on Sobolev spaces, difference quotients are defined in section 5.8.2. Then follows theorem 3:
(i) Suppose $1\leq p < \infty$ and $u\in W^{1,p}(U)$. Then for each $V\subset\subset U$ $$||D^hu||_{L^p(V)}\leq C||Du||_{L^p(U)}$$ for some constant $C$ and all $0<|h|<\frac{1}{2}$dist$(V,\partial U)$.
My question is the following: is it really necessary for $h$ to lie between $0$ and $\frac{1}{2}$dist$(V,\partial U)$? Why is there a need for the coefficient $\frac{1}{2}$? I read through the proof of the theorem, but I couldn't see why $h$ couldn't be larger than $\frac{1}{2}$dist$(V,\partial U)$; for instance if $h=\frac{3}{4}$dist$(V,\partial U)$ I could see no problems arising.
I would guess that you don't need it to be $\tfrac{1}{2}$. But presumably you wouldn't rather the theorem said:
"Fix $\alpha \in (0,1)$"....
and then
"for all $0 < |h| < \alpha\;\mathrm{dist}(V,\partial U)$".
This would just look confusing and you'd wonder what the relevance of $\alpha$ really was. Analysts do things like this all the time.