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If I have a bounded and homogeneous function of degree $-1$, can I conclude that it goes to $0$ pointwise at infinity, isn't it?

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Defined on which set? –  Did Mar 1 '13 at 17:27
    
In a subset of $\mathbb{R}^n$ –  Mario Mar 1 '13 at 17:28

1 Answer 1

If $f$ is homogeneous of degree $-1$ then $$ f(t\,x)=\frac1t\,f(x),\quad t>0,\quad x\ne0. $$ Choose $t=1/|x|$ to get $$ f(x)=\frac{1}{|x|}f\Bigl(\frac{x}{|x|}\Bigr). $$ Let $M$ be an uper bound for $f$. Then $$ |f(x)|\le\frac{M}{|x|},\quad x\ne0. $$

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Thank you very much! –  Mario Mar 1 '13 at 17:35

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