IR-TST 2012 Problem 5

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$ :a) $f(a)=0 \Leftrightarrow a=0$ b) $f(ab)=f(a)f(b)$ c) $f(a+b)\le 2 \max \{f(a),f(b)\}$ .

Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$ .

Proposed by Masoud Shafaei