# Homogeneous function of degree $-1$

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?

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

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 $$|f(x)|\le\frac{M}{|x|},\quad x\ne0.$$