Let $c\in E\subseteq \Bbb R$, $f:E\to\Bbb R$. Suppose $k\in\Bbb N$ and $k\geq 2$, and $f$ is $k$-differentiable at $c$ (i.e., $f$ is $(k-1)$-th differentiable on a neighborhood of $c$, and $k$-th differentiable at $c$.) And suppose $f$ is not infinitely differentiable at $c$.
Theoretically, there is nothing suprising that such $f$ exists. However, in practice, do we actually have some chance to see or to deal with such functions? For example, $\sin,\cos, e^x$, polynomials, power functions, ...etc, are infinitely differentiable, and the functions made up by using them with operations($+,-,\times,/,\circ$, etc) are usually good enough. And even those functions that are not infinitely differentiable, are frequenly "too bad" that even not being "twice-differentiable" or "differentiable". So that's where my question arises, is it common to see a function, not so bad, which is twice(or higher, say $3$rd, $4$th, ...)-differentiable, but even not so good, that cannot be infinitely differentiable?