I am trying to prove the following proposition:
Proposition. Given a unimodal probability distribution $f(u)$, symmetric around $u=0$, strictly increasing for u<0 and strictly decreasing for $u>0$, then for all $y \ge 0$, $ \int_{x-y}^x f(u)du - \int_x^{x+y} f(u)du = 0 \hspace{0.1in} \Leftrightarrow \hspace{0.1in} x=0.$
The proof of $\Leftarrow$ is simple (plug in $x=0$), but the proof of $\Rightarrow$ is escaping me. Even though it feels very intuitive, I can't seem to nail it down rigorously. Is it true?
(In terms of strategy, I'm trying is to break it up in three cases: $x>0$, $x=0$, x<0, and try and find a contradiction for the two $x \ne 0$ cases, but I'm not getting it.)
(Assumptions about continuous differentiability of $f$ are fine if necessary.)