Mallat in his book on wavelets makes the following definition - A function $f : [a,b] \to \mathbb{R}$ is $(C, \alpha)$-Lipschitz at $v \in [a, b]$ if there is a polynomial $p_v$ of degree at most $\lfloor \alpha \rfloor$ such that $|f(x) - p_v(x)| \leq C|x - v|^\alpha$ for any $x \in [a, b]$. It is uniformly Lipschitz if it is Lipschitz for all $v \in [a ,b]$ with a constant that is independent of $v$. Continuous differentiability implies Lipschitz continuity as above, since one can use the Taylor polynomial of the function.
Question: Apparently, the converse is also true - uniform $(C, \alpha)$-Lipschitz continuity implies that the function is $\lfloor \alpha \rfloor$ times continuously differentiable. I'm having difficulty proving this. It is clear that if $\alpha > 1$ then the function is differentiable. I'm guessing the derivative is uniformly $\alpha-1$ Lipschitz which would complete the proof by induction, but I'm having trouble showing this. Any hints would be appreciated!