In real analysis, it seems that for every nice condition you might expect to hold, there exist numerous counterexamples, often with creative constructions. For example, the Weirstrass function shows continuous does not imply differentiable, and Volterra's function is a function with a bounded derivative which is not Riemann integrable.
Often (especially in physics), we do away with these complications by assuming the functions we are working with are smooth; that is, that they are infinitely differentiable. So, I am asking whether there are any examples of smooth functions which one might consider pathological, or which have some unusual or unexpected properties.
The best example I can think of is smooth functions which are not analytic. In particular, one can construct functions like$$ F(x) = \sum_{k\in \mathbb{N}}e^{-\sqrt{2^k}}\cos(2^k x)$$Which are smooth everywhere but analytic nowhere in $\mathbb R$. Are there any other properties one might expect smooth functions to have, which they do not?