I have a homogeneous of degree $s$ function $f : \mathbb R^n \setminus 0 \to \mathbb R$ which is of class $C^k$. Extend $f$ to a function $\tilde f : \mathbb R^n \to \mathbb R$ given by setting $\tilde f (0) := 0$. I am trying to show that $\tilde f$ is also of class $C^k$ on $\mathbb R^n$, given that $s > k$.
I have shown that $D^\alpha f := \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} f$ is homogeneous of degree $s - | \alpha| $, where $|\alpha| \leq k$. To show $\tilde f$ is $C^k$ on $\mathbb R^n$, I need to show that all of the partial derivatives (i.e. $D^\alpha f$ for all multi-indices $\alpha$) exist and are continuous at all points of $\mathbb R^n$. Since $\tilde f$ is just $f$ away from $0$, the only point we need to worry about is $0$.
For existence, I suppose I could use the limit definition of the partial derivative, but since $\alpha$ is a multi-index this would seem overly complicated. I feel there must be a simpler way using the homogeneity fact I have shown above. For continuity, I'm not sure how to proceed because I don't have an explicit form for the partial derivative.