Read the first two lines of this section
I think it is sufficient to say that $f$ is $(k+1)$-times differentiable in an interval $I$ containing $a$ with $$q \leq |f^{(k+1)}(x)| \leq Q$$ for all $x \in I$.
$f^{(k+1)}(x)$ does not have to be continuous. Anyone has thoughts on this.