The exact question I'm faced with is
Let f be a continuous real-valued function on $0 < x < \pi$ such that $f(0)=f(\pi)=0$ and $f'\in L^2(0,\pi)$. For which functions do we have $\int_0^{\pi}|f(x)|^2dx=\int_0^{\pi}|f'(x)|^2dx$?
I know that convergence in $L^2(0,\pi)$ means that the functions $f$ and $f'$ can still differ on a zero set of points. I still can't think of what type of function is equal to its derivative except for a few points, or on the whole interval for that matter. The only function I can think of is $e^x$, but I was wondering if there is a more general property that ensures the equality above.