Two variable function on a circle.

Question

I've been preparing for my exams and encountered this exercise that I can't figure out: Let f be a two variable, continuous function on a circle $K$ with the center $({x}_0, {y}_0)$. $\iint_R f dR$ is equal to 0 for any rectangle $ R \subset K$ that contains point $({x}_0, {y}_0)$. Prove that $f({x}_0, {y}_0)=0$. My intuition: For any arbitrary rectangle, which contains $({x}_0, {y}_0)$ we can find smaller one, that will also include this specific point. If we take infinitely small rectangle and compute the limit of function's value with respect to rectangle's diagonal we will probably get that $f({x}_0, {y}_0)=0$. This is only my intuition, but now, how do I write a sufficient proof?