Homogeneous function of degree $-1$

Question

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?

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