Putnam 1977 A6

Problem A-6 Let $f(x, y)$ be a continuous function on the square $$S=\{(x, y): 0<x<1,0<y<1\} .$$

For each point $(a, b)$ in the interior of $S$, let $S_{(a, b)}$ be the largest square that is contained in $S$, is centered at $(a, b)$, and has sides parallel to those of $S$. If the double integral $\iint f(x, y) d x d y$ is zero when taken over each square $S_{(a, b)}$ must $f(x, y)$ be identically zero on $S$ ?