Is a function concave if and only if $f(x)/x$ decreasing?

Question

I'm trying to prove the following (which is useful for my non-math research): Suppose that a function $f:[0,1] \to [0,1]$ is non-decreasing and $f(0)=0$. Then $f$ is concave if $f(x)/x$ is non-increasing. I'd like to show this without imposing differentiability on $f$. Would anyone have a suggestion? Thank you:)

Answer 1

Consider $$f(x)=\begin{cases} 2x & 0 \leq x < \tfrac13 \\[1ex] \tfrac23 & \tfrac13 \leq x < \tfrac23 \\[1ex] x & \tfrac23 \leq x \leq 1 \end{cases} $$