Does continuous and strictly increasing implies convex function?

Question

Let $f:[0,\infty)\to [0,\infty)$ be a continous and strictly monotone increasing function and $f(0)=0$. Then prove or disprove that $f$ is a convex function.

My initial guess that, $f$ is a convex function, I want to prove it.

I am unable to proceed!I'm not getting any idea how to use continuity. Any hint?

Answer 1

Take $$ f(x) = \log(1+x) $$ Then $f(0)=0$, $f'(x) = 1/(1+x)>0$, but $f''<0$, so $f$ is concave but satisfies all the hypotheses.