Lp spaces examples

Question

Please i need some help with this exercises Let $f \in L^p(\mathbb{R})$.

Prove that

$\lim_{t\rightarrow 0} \int_{\mathbb{R}}|f(x+t)-f(x)|^p dx =0$

And i have this hint:

Prove that $C(\mathbb{R}) \cap L^{p}(\mathbb{R})$ is dense in $L^p(\mathbb{R})$ , then show the results using the fact that $f$ is continuous

Thak's !!

Answer 1

Convolute $f$ with a mollifier to get a $C(\Bbb R) \cap L^p(\Bbb R)$ sequence converging to $f$ in $L^p(\Bbb R)$. Next, prove the desired result for $C(\Bbb R) \cap L^p(\Bbb R)$ functions. Finally, do an approximation argument. Let me know if you need further hints.