Suppose I have $f \in L^1\cap L^\infty$, and I want to take the following limit $$\lim_{h\to0} \int|f(x+h)-f(x)|dx$$
Does this follow from the dominated convergence theorem? Since $|f(x+h)-f(x)| \leq |f(x+h)|+|f(x)|$, and $\|f(x+h)\|_{L^1(\mathbb{R})}<\infty$ and $\|f(x)\|_{L^1(\mathbb{R})}<\infty$, we then have $$\lim_{h\to0} \int|f(x+h)-f(x)|dx = \int\lim_{h\to0}|f(x+h)-f(x)|dx = 0$$?
I'm not convinced, and I think a better argument would be to involve the density of continuous function in $L^1$. If the approach in this post does not work, can anyone explain why?