Suppose we have a continuous function $f: (0,1) \to (0,1)$. Does there exist a differentiable function $\phi: (0,1) \to (0,1)$ such that $f(x) \leq \phi(x)$?
There does exist such a differentiable function if the function $f$ has finitely many non-differentiable points. Also this can be done if there exists an $\epsilon >0$, no matter how small, such that the function $f$ is differentiable in $(0,\epsilon)$.
Note that the real challenge lies in finding the differentiable function $\phi$ to be strictly less than one at the same time. This seems very much possible, at least pictorially. Because the graph of the function $f$ lies strictly below the $Y=1$ line. And hence one can draw a smooth curve between the graph of $f$ and the $Y=1$ line. But I can not prove it mathematically (rigorously). I need your help. Thanks in advance!