KöMaL 2007 A420

Find all functions $f\colon (0,\infty)\to\mathbb{R}$ such that | $f$($x$)-$f$($y$) | $\le$|$x$-$y$| for all $x$,$y$>0 and there exist positive constants $a$, $b$, $c$ satisfying $f(ax)+f\left(\frac{b}{x}\right) =f(cx)$ for all $x$.