Differentiability class: Example of maps that are $C^k$ but not $C^{k+1}$. Is there a classical example for the fact that the differentiability class satify $$ C^{k+1} \subsetneq C^{k} $$ I'm interested in the $C^{k+1} \neq C^{k}$, then is I'm looking for a **classical** > Example of maps that are $C^k$ but not $C^{k+1}$. Maps $f_j:\mathbb{R}\to\mathbb{R}$. For a compact interval the following example should work $$ f_k:[0,2]\to\mathbb{R},\quad f(x)=\begin{cases} 0&\text{for $x\in[0,1)$}\\\ 1&\text{for $x\in[1,2]$} \end{cases} $$ then I build recursively $f_{k-1}$ by integrating $f_k$ thus the obtained $f_0$ or $f_1$ is $C^k$ but not $C^{k+1}$.

Is the Weierstrass function classical enough? It is **_continuous but nowhere differentiable_** ; if we integrate it $k$ times we obtain a function which is $C^k$ but not $C^{k + 1}$ for any point in its domain!