Find all monotonic functions $f: R^+ \rightarrow R^+$ such that :
$f(x+y) \le f(x) + f(y)$
$f(0)=0$
Background: This exercise was created by me asking myself if, given a distance function $d_1(x,y)$, a function $d_2(x,y)=f(d_1(x,y))$ could still be a proper distance for some function $f$. If such a function $f$ exists than $d_2$ would be a proper distance because:
$d_2(x,z)=f(d_1(x,z))$
$\le f(d_1(x,y)+d_1(y,z))$ ( monotonicity of f+ triangular for $d_1$)
$\le f(d_1(x,y))+f(d_1(y,z))$ (property of f)
$=d_2(x,y)+d_2(y,z)$