97832 – We Solve Problem

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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$)$.

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