Let $f: \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ be a continuous differentiable and homgenuous function of degree $\alpha$. First we had to show with use of the definition of the partial derivative that the $i$th partial derivative $D_{i}f$ is homogenous of degree $\alpha - 1$. I managed to do this, but I am struggling with the second question.
Show that we can expand $f$ to a continuous differentiable function on $\mathbb{R}^{n}$ if $\alpha > 1$. The question also refers to a proposition in the syllabus which says for a function $f: \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ non-constant, continuous and homogenuous of degree $\alpha$, then $\lim_{x\to 0}f(x)$ is equal to $0$ if $\alpha > 1$ and does not exist if $\alpha \leq 0$.
I understand that the function $f$ can be expanded to a continuous function and I assume that the proof to this question involves a similar explanation but with respect to the partial derivates of $f$ yet I'm still stuck on how to prove this. Thanks in advance!