(I've been lighting this board up recently; sorry about that. I've been thinking about a lot of things, and my professors all generally have better things to do or are out of town.)(adsbygoogle = window.adsbygoogle || []).push({});

Is there an easy way to show that if [itex]f[/itex] is Lipschitz (on all of [itex]\mathbb R[/itex]), then

[tex]

\int_{-\infty}^\infty f^2(x) e^{-\frac{x^2}{2t}}dx < \infty \quad ?

[/tex]

I've tried a number of different approaches - boundedness of the derivative and integration by parts, bounded/dominated/monotone convergence theorem, approximation of [itex]f^2[/itex] by polynomials - but I haven't gotten any of them to work. Apparently, just because [itex]f[/itex] is Lipschitz doesn't mean [itex]f^2[/itex] is Lipschitz. (Just look at [itex]f(x) = x[/itex].)

Can anyone think of a Lipschitz function for which the integral above is infinite?

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# Integrability and Lipschitz continuity

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