Suppose $1\leq p < \infty$, and that $\mathbb{R}^d$ is equipped with lebesgue measurea. Show that if $f\in L^p (\mathbb{R}^d)$, then
$||f(x+h)-f(x)||_{L^p}\to 0$ as $|h|\to 0$.
I want to use dominated convergence theorem. First note that the question is equivalent to $||f(x+1/n)-f(x)||_{L^p}\to 0$ as $n\to \infty$ and $n\to -\infty$.
But we see that $||f(x+1/n)-f(x)||_{L^p}\leq (\int |f(x+1/n)|^p+|f(x)|^p)^{1/p}=(\int (2|f(x)|^p)^{1/p}<\infty$. So we can use dominated convergence theorem to conclude. But the hint provided in this problem suggests that this argument is naive and wrong. What did I do wrong?
My guess is that since $f$ is not continuous we cannot conclude $f(x+h)-f(x)\to 0$. Is this the only part that I missed?