$n(fx)=f(nx)$ for natural number $n$ and $f(x)$ is an increasing real function. So $f$ is linear?

It is known that, for real increasing function $f:[0,1]\to\mathbb R$, if $f(ax)=af(x)$, then $f$is linear.

Now consider $a=n$ where $n$ is natural.

Is it possible that $f(x)$ is not linear on a positive measure of points?

I stuck here: say $f(x)=cx$ on the rationals, then, how to prove that $f(x)=cx$ on all reals?