1. The problem statement, all variables and given/known data My question is similar to this one (https://www.physicsforums.com/threads/metric-function-composed-with-concave-function.566338/), I think. I have a concave function [itex]f: [0, \infty) \rightarrow [0, \infty)[/itex] where [itex]f(0) = 0, f(x) > 0, x \neq 0 [/itex] and I need to show that this function is always increasing, or at least non-decreasing. 2. Relevant equations 3. The attempt at a solution I'm trying a proof by contradiction. If the function isn't increasing, then [itex] f(a) > f(b)[/itex] for some [itex]a < b[/itex]. That gives me [itex]f(a) - f(b) > 0[/itex], but I don't see how to proceed. I know that [itex]f(a + b) \leq f(a) + f(b)[/itex], i.e. the function is subadditive, but once again, I'm stuck.