If $f\in L^p(\mathbb{R^n})$ $1\leq p < \infty$ then prove $\displaystyle\lim_{|h|\to 0} \|f(x + h) - f(x)\|_p = 0$.
I had tried to prove the continuity a.e. of $\|f(x + h) - f(x)\|_p$, then could pass the $\lim$. This way is good or there is other more simple?
On
Its easier to prove it first for $C^\infty_c$. For these functions we have mean value theorem(for $\Bbb R^n$ its only an inequality), valid for every $x,h$ $$|f(x+h) - f(x)| ≤ |\nabla f(x_0)h| ≤ ‖∇ f‖_∞ |h|$$ So if $\text{supp }f = U$, $$ ∫_{\mathbb R^n} |f(x) - f(x+h)|^p \ \text dx ≤ \mu(U )‖ ∇ f‖^p_∞ |h|^p → 0 $$ Then conclude by density; choose $f_n$ smooth compactly supported with $f_n → f ∈ L^p$, and $n$ large we obtain $$‖ f(x+h) - f(x) ‖_p ≤ ‖ f(x+h)- f_n(x+h)‖_p + ‖ f_n(x+h)- f_n(x)‖_p + ‖ f(x)- f_n(x)‖_p ≤ 3h → 0 $$
Hint: Since $f$ is not continuous, I believe you will have to use the "standard framework" to prove this. Specifically, I mean that you should first establish the claim for simple functions $\varphi$ and then use this to establish the claim for more general functions.