Show that a strict positive homogeneous function of degree 0 has a minimum and maximum.

Question

I'm stuck with the following question:
"Let $ f : \mathbb R^k \setminus {(0,0,...,0)} \rightarrow \mathbb R$ be a twice continuously differentiable and strictly positive homogeneous of degree 0. prove that it has a minimum and maximum at $ \mathbb R^k \ \setminus {(0,0,...,0)} $."
I've already shown that the Hessian of the function is neither positive definite nor negative definite. I know that from Euler's identity, we have $ \langle\nabla f(a) , a \rangle = 0 $ for all $ a \in \mathbb R^k \setminus {(0,0,...,0)} $, however, I don't know how to show that it has a minimum or maximum.
I thought maybe I'll show a point that has a minimal/maximal value, however, I don't really have a point of thought about that.

Answer 1

First, homogeneous of degree 0 means $f(x)=f(x/\|x\|)$ with $x/\|x\|$ on the unit sphere $S^{k-1}=\{x\in\mathbb R^k|\|x\|=1\}$. Hence it suffices to restrict $f$ to this sphere without changing its image. But it's compact, so its image by any continuous function is also compact, thus $f$ attains maximum and minimum on this sphere (and thus all of $\mathbb R^k$).