# A smooth nowhere analytic function such that all derivatives are monotone

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?