Infinitely differentiable How can one find if a function $f$ is infinitely differentiable?

for functions:
Each analytic function is infinitely differentiable.
Each polynomial function is analytic.
Each Elementary function is analytic almost everywhere. I assume this is valid also for the Liouvillian functions.
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for function terms:
The set of the function terms of the Elementary functions is closed regarding differentiation.
The set of the function terms of the Liouvillian functions is closed regarding differentiation.
If $n_1,n_2\in\mathbb{N}_0$, $n_1\ne n_2$, $f\colon z\mapsto f(z)$, and
$$\frac{d^{n_1}}{dz^{n_1}}f(z)=\frac{d^{n_2}}{dz^{n_2}}f(z),$$
then $f(z)$ is infinitely often differentiable.
There are some general differentiation rules for calculating $n$-th derivatives, e.g. higher factor rule, higher sum rule, higher product rule, higher chain rule.
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A function with an infinitely differentiable function term is infinitely differentiable if each of its $n$-th derivatives is differentiable.
