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TOT 063 1984 Spring S-O4 S-A4 f:[0,1] \to [0,1], graph covered by n rectangles

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August 19, 2019
functiongraphRectanglescombinatoricscombinatorial geometrygeometryrectangle

Problem Statement

Prove that, for any natural number nn, the graph of any increasing function f:[0,1]ā†’[0,1]f : [0,1] \to [0, 1] can be covered by nn rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points. (a) Assume that f(x)f(x) is continuous on [0,1][0,1]. (b) Do not assume that f(x)f(x) is continuous on [0,1][0,1].
(A Andjans, Riga)
PS. (a) for O Level, (b) for A Level
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