Why does a function having any 'kinkiness' not infinitely differentiable?
Some of the functions(e.g. $f(x) = (|x|)^{k}$, $k \in \mathbb{N}$) belong to the class of $C^{k-1}$ functions(i.e. for these functions $f'(x)$, $f''(x)$, $...$ , $f^{k-1}(x)$ exist and are continuous). They are able to evade up to $(k-1)^{th}$ differentiation. What is inherent in the idea of differentiation and its continuity which is able to capture any 'kinkiness' behavior of the function?