Same integral on every interval of fixed length

Question

Let $f$ be a continuous function defined on some interval $I \subseteq \mathbb R$, and let $\ell$ be a positive real number. It seems pretty intuitive to me that, if $\int_a^b f$ is the same for every $a$ and $b$ in $I$ such that $b-a=\ell$, then $f$ is a constant function. How can we prove this, though?

Answer 1

This is false. If $f$ is any periodic function with period $p$, then $\int_a^{b} f(x)dx$ is the same for all intervals $[a,b]$ of length $p$. In particular you can take $f(x)=\sin x$ and $\ell=2\pi$.