The other day (for purpose of modelling temporary "wobbles") I investigated
$$f(t): t\to \exp\left(\frac{N\pi i}{N\pi|t|+1} \right), \forall t\in \mathbb R, N\in \mathbb N$$
I suppose it shall be trivial to prove it is differentiable everywhere infinitely many times except $t=0$. But how to show it for $t=0$? Or maybe it isn't differentiable there? If so, can we use some (algebraic or numerical) technique to get it differentiable there?
I suppose the closely related function:
$$g(t): t\to \exp\left(\frac{N\pi i}{N\pi t^2+1} \right), \forall t\in \mathbb R, N\in \mathbb N$$
Is nicer in this regard? As far as I can tell all of it's derivatives shall follow $$g^{(k)}(t) = Q(t) \exp\left(\frac{N\pi i}{N\pi t^2+1} \right)$$
For some rational function $Q_k(t) = \frac{P_{k1}(t)}{P_{k2}(t)}$ for some sequence of polynomials $P_{k1},P_{k2}$.
Only remains to prove that $P_{k2}$ has no zeros on real line for any $k$?
Consider
$$\lim_{t\to 0^+}\frac{f(t)-f(0)}{t}.$$
Verify that this limit, which is the derivative of $f$ from the right at $0,$ exists and is a nonzero number $L.$ Now note that $f$ is even. Therefore the derivative of $f$ from the left is $-L\ne L.$ It follows that $f'(0)$ does not exist.
As for $g,$ it is the composition of $x\to e^{ix}$ with $t\to \dfrac{N\pi}{N\pi t^2+1}.$ Both of these functions are in $C^\infty(\mathbb R),$ hence so is $g.$ (No computations necessary.)