If $f \in L^p(\mathbb{R})$, where $p\in [1, \infty)$, then $\int_{-\infty}^\infty |f(x+y)-f(x)|^p dx \to 0 \ \textrm {as} \ \ y \to 0.$
Assume now that a function $f: \mathbb{R} \to \mathbb{R}$ is locally in $L^p$, that is for each compact interval $[a,b]\subset \mathbb{R}$ we have $f \in L^p[a,b]$. I would like to know is it then $\int_{[a,b]} |f(x+y)-f(x)|^p dx \to 0 \ \ \text{as} \ \ y \to 0.$