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I am studying $L^p$ spaces and I ve found this exercise I could not solve.

Let $f$ $L^p(\mathbb{R}^n), 1 \le p < \infty$ then $\|f(x + h) - f(x)\|_{L^p} \to 0 \Leftarrow |h| \to 0.$

It supposed to be easy since the continuos functions with compact support are dense in $L^p(\mathbb{R}^n)$ but I don't how to use this to conclude the result.

I tried to do:

Given $f \in L^p(\mathbb{R}^n)$ there existis a sequence $(f_n) \in \mathcal{C}_c(\mathbb{R}^n)$ such that $\|f_n - f\|_{L^p} \to 0 \Leftarrow n\to \infty.$

$\|f(x + h) - f(x)\|_{L^p} \le \|f(x+h)-f_n(x)\|_{L^P} + \|f(x) - f_n(x)\|_{L^p}.$

So, I can work with the second right term, but I have no idea what to do with the first.

Thanks

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    See here: http://math.stackexchange.com/questions/1802705/let-f-in-lp0-1-and-define-f-h – 2017-01-06

1 Answers 1

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I suppose you mean $(f_n) \subset C_c(\Bbb R^n)$. Hint:

$$f(x+h) - f_n(x) = f(x+h) -f_n(x+h) + f_n(x+h) - f_n(x)$$

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    Thank you. Its done. The first therm that comes from your suggestion will be limited by the same argument as I stated on my question, for the second I must use uniform continuity. It was quite obvious. Thanks. – 2017-01-06