I have a question about remark 7 which is written after Propostion 9.3 in Brezis Functional Analysis book. It states in particular that if $f\in W^{1,\infty}(\Omega)$, then there exists $C>0$ for every $K \subset \Omega$ that is compact, for all $h\in \mathbb R^n$ such as $|h|<\text{dist}(K,\partial\Omega)$ we have $$\Vert\tau_hf-f\Vert_{L^\infty(K)}\le \| \nabla f\|_{L^\infty(\Omega)} |h|$$ where $\tau_hf(x)=f(x+h)$.
Remark 7 states that this proposition implies that every $f\in W^{1,\infty}(\Omega)$ has a continuous representative.
Can somebody explain me how to obtain the last statement by using only propostion 9.3 if it is possible?
In fact, the proposition implies that for every $K \subset \Omega$ that is compact, for every $h\in \mathbb R^n$ such as $|h|<\text{dist}(K,\partial\Omega)$ and for ALMOST EVERY $x \in K$ we have $$|f(x-h)-f(x)| \leq |h| \| \nabla f\|_{L^\infty(\Omega)}$$
And that seems to be not enough to prove that $f$ has a continuous representative, because it is not correct to say that for almost every $x \in K$ and for every $h\in \mathbb R^n$ such as $|h|<\text{dist}(K,\partial\Omega)$ we have $$|f(x-h)-f(x)| \leq |h| \| \nabla f\|_{L^\infty(\Omega)}$$
I have also remarked that if we prove that $f$ is continuous almost everywhere, it is easy to deduce that $f$ has a continuous representative. However, I haven't managed to prove that $f$ is continuous almost everywhere.
Feel free to give some hints in the comments, to give some advice or to ask some intermediate questions. Any comment is welcome!