EDB — 192

[192] Let $f:[0,\infty )\to ℝ$ be concave, with $f(0)=0$ and $f$ continuous in zero.

• Prove that $f$ is subadditive, i.e.

  $$f(t)+f(s)\ge f(t+s)$$

  for every $t,s\ge 0$. If moreover $f$ is strictly concave and $t>0$ then

  $$f(t)+f(s)>f(t+s).$$

• Prove that, if $\forall x,f(x)\ge 0$, then $f$ is weakly increasing.

• The other way around? Find an example of $f:[0,\infty )\to [0,\infty )$ with $f(0)=0$, continuous, monotonic increasing and subadditive, but not concave.