Assume $f$ is integrable on $[0,\infty)$ and assume that for $a,b>0$, the value of $\int_{a}^{ab} f(x) dx$ is independent of $a$.
Prove that $f(x)=\frac{c}{x}$, where $c$ is a constant.
I have tried several things, like showing $g(x)=xf(x)$ must have derivative zero, but I am unsure how to use the integral assumption.
Thanks!