Characterizing bell-shaped curves

I tried to characterize mathematically what makes a function bell-shaped. I found the following:

Definition. A $C^\infty$-function $f:\Bbb R\to\Bbb R$ is called bell-shaped if its $n$-th derivative has exactly $n$ zeros (counted with multiplicity) for all $n\in\Bbb N_0$.

I do not claim that any intuitively bell-shaped curve is included (e.g. non-smooth functions), but I felt pretty confident that there are no intuitively non-bell-shaped curves in this class. However, my imagination is limited, so I want to ask whether anyone can find a pathological example of some curve in this class for which it might be debatable to call it bell-shaped in an intuitive sense. Upside down or asymmetric bells are okay for me.

Also, is it true that any such function must be bounded? I would consider an unbounded function as a counterexample. Then the next question would be whether it suffices to extend the definition by the predicate “bounded”.

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