On integrals constant on congruent domains

We prove that if a real function of two variables is defined, continuous, and bounded on the whole plane, then it is constant under the condition that its integral on each square of unit area is constant. We point out variants of this theorem. We present an example of a function that is not constant but whose integral on each circle of unit radius is constant. Such a function is $\sin\beta x$, where $P$ is any root of the Bessel function $J_1$.