KöMaL 2012 A565

The positive integers are coloured with a finite number of colours. A function $f$from the set of positive integers to itself has the following two properties: ($a$) if $x \le y$, then $f$($x$)$\le f$($y$); and ($b$) if $x$, $y$, and $z$ are (not necessarily distinct) positive integers of the same colour and $x$+$y$=$z$, then $f$($x$)+$f$($y$)=$f$($z$). Does it follow that the function $\frac{f(x)}x$ is bounded from above?