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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 !!

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    Related: [exact same question](https://math.stackexchange.com/questions/2098605/show-lim-t-to-0-int-bbb-rn-mid-fxt-fx-mid-p-d-mu-n-x) and [this one answered](https://math.stackexchange.com/questions/1019483/show-that-lim-t-to-0-int-mathbbrdfx-fx-tdx-0) – 2017-01-15

1 Answers 1

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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.