MathDB
image as a subset of rectangles

Source: VJIMC 2010 2.4

June 4, 2021
functiongeometryrectangle

Problem Statement

Let f:[0,1]→Rf:[0,1]\to\mathbb R be a function satisfying ∣f(x)−f(y)∣≤∣x−y∣|f(x)-f(y)|\le|x-y|for every x,y∈[0,1]x,y\in[0,1]. Show that for every ε>0\varepsilon>0 there exists a countable family of rectangles (Ri)(R_i) of dimensions ai×bia_i\times b_i, ai≤bia_i\le b_i in the plane such that {(x,f(x)):x∈[0,1]}⊂⋃iRi and ∑iai<ε.\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
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