Trouble understanding tiling board with tiles of at least one integer dimension. I am trying to understand this proof (problem 2 pages 1-2) which shows that if a rectangle can be tiled by smaller rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. I can't seem to understand why "the integral of $f$ over any horizontal or vertical line segment with integer length is zero.". How can this be proven mathematically? Could someone help me with the proof linked?

It means that $f(x,y)=\exp(2\pi i(x+y))$ has the property that $$\int_{a}^{a+m}f(x,y)\,dx=0$$ for any $a$, $y$ and integer $m$, and $$\int_{b}^{b+n}f(x,y)\,dy=0$$ for any $b$, $x$ and integer $n$.

The key to the proof is that $$\int_a^b\int_c^d f(x,y)\,dx\,dy=0$$ if and only if at least one of $b-a$ and $d-c$ is an integer.