n Times differential functions
Most easy examples of functions are $\mathscr{C}^\infty$, that is they are infinitely often differentiable, like $\sin$ or $\exp$ or polynomials. A rather cute, if trivial, construction of functions that are not is the anti-derivative of the absolute value $|x|$.
$$ \int_0^{x} |\tilde{x}|\,d\tilde{x} = \begin{cases} -\frac{1}{2} x^2 & \text{if $x < 0$} \\ \frac{1}{2} x^2 & \text{if $x \geq 0$} \end{cases} $$
and generalizing to functions $f_n \in \mathscr{C}^n$ but not in $\mathscr{C}^{n+1}$ is
$$ f_n = \begin{cases} - x^n & \text{if $x<0$} \\ x^n & \text{if $x \geq 0$} \end{cases} $$
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