If $f$ is a continious function from $\mathbb{R}$ to $\mathbb{R}$ and there exists $C$ such that:
$\forall x\in\mathbb{R} :\int_x^{x+T}f(t)dt=C$
then $f$ is $T$-periodic.
Proving the inverse is easier and has already been answer on another question.
2026-03-30 13:21:36.1774876896
If $\int_x^{x+T}f(t)dt=C$ then $f$ is periodic
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HINT:
Differentiate both sides of the equation $F(x)=\int_x^{x+T}f(t)\,dt=C$ with respect to $x$. What can you immediately conclude?