Do we have $C^{\infty}\subsetneq C^k\subsetneq ...\subsetneq C^1\subsetneq C^0\ \forall k\in\Bbb N$?
Is it possible to find for each $k$ some function $f_k:\Bbb R \mapsto \Bbb R$ such that $f_k\in C^k$ but $f_k\notin C^{k+1}$
$\mid x^k\mid\in C^{k-1}$ and $\notin C^{k}$ works when $k$ is odd. Is there an example for $k$ even?