I have the following problem. Let $\rho(s)\geq 0$ for all $s\geq 0$ be a continuous concave function, satisfying $\rho(s)\geq\rho_1(s)$ for all $s>0$, where $\rho_1$ is a strictly increasing and positive function (not necessarily continuous). Does it follow that $\rho$ is also strictly increasing?
My guess is that this is the case, otherwise there would exist $s'>0$ such that $\rho_1(s)>\rho(s)$, arriving at a contradiction. However, the existence of this $s'$ is not evident to me. If anyone could guide me towards an answer, I would greatly appreciate it.