So the problem is if: $$\int_0^x f(t)\, dt = f(x) $$ then f(x) is identically zero.
So far I've tried an approach with the mean value theorem and I end up with the equation: $$f(x) = xf(a)$$ for some $a$ in $[0, x]$ for all $x$.
And that's as far as I got. I think the mean value theorem is the right approach to this, but I don't know what to do much after that. I also think an approach would be Riemann sums, but I didn't get too far with that either. So any ideas?