Related questions that might provide some context: (1) (2) (3) (4)
Let's restrict our attention to real-valued functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples $\!^{[1]}$$\!^{[2]}$ of smooth (class $C^\infty$) functions that are nowhere real-analytic. Is there a smooth nowhere analytic function such that the function itself and its derivatives of any order are monotone?
The answer is negative. All such functions are analytic. For totally monotone functions it was proved by Bernstein, and for the general case see J. A. M. McHugh "A proof of Bernstein's theorem on regularly monotonic functions".