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Is it true that for any real valued Borel measurable square integrable function $f$, ${\displaystyle\lim_{s\rightarrow 0}\int\limits_{\mathbb{R}}\left(f(t-s)-f(t)\right)^2\,dt=0}$ ? If yes, then how?

Partha Pratim Ghosh
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  • That is usually done by exploiting the density of $C^{\infty}$ in $L^2$: http://math.stackexchange.com/questions/1018716/translation-operator-and-continuity – Jack D'Aurizio Feb 10 '17 at 13:14
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    In $L^2$ that is even easier since we may switch to Fourier transforms and simply study the behaviour of $(e^{i s\xi}-1)$ close to $\xi=0$. – Jack D'Aurizio Feb 10 '17 at 13:48

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