Convergence in $p$-norm of $f(x+h)-f(x)$ as $h$ tends to zero.

Question

I'm working on the following problem from a real analysis qualifying exam from Texas A&M from Spring 2001, and I'm not sure how to proceed:

Suppose $f\in L^p(\mathbb{R})$ and $1\leq p<\infty$. Prove that $\lim_{h\to 0}\int_{\mathbb{R}}|f(x+h)-f(x)|^p\,dx=0$.

Answer 1

This is a very standard result in real analysis done by so called "density argument". One should be able to check it in any advanced real analysis textbook. See for instance Folland's Real Analysis:

Or Tao's excellent note (Proposition 19) regarding the density argument.