Problem #PRU-97832

Problem

$f(x)$ is an increasing function defined on the interval $[0,1]$. It is known that the range of its values belongs to the interval $[0,1]$. Prove that, for any natural $N$, the graph of the function can be covered by $N$ rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. (In a rectangle we include its interior points and the points of its boundary).