EDB — 192

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

• Prove that \(f\) is subadditive, i.e.

  \[ f(t)+f(s)≥ f(t+s) \]

  for every \(t,s≥ 0\). If moreover \(f\) is strictly concave and \(t{\gt}0\) then

  \[ f(t)+f(s){\gt} f(t+s)~ . \]

• Prove that, if \(∀ x, f(x)≥ 0\), then \(f\) is weakly increasing.

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