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Uncountably much solutions

Source: IMC 2002 day 1 problem 5

October 7, 2005
functionreal analysisreal analysis solved

Problem Statement

Prove or disprove the following statements: (a) There exists a monotone function f:[0,1]→[0,1]f : [0, 1] \rightarrow [0, 1] such that for each y∈[0,1]y \in [0, 1] the equation f(x)=yf(x) = y has uncountably many solutions xx. (b) There exists a continuously differentiable function f:[0,1]→[0,1]f : [0, 1] \rightarrow [0, 1] such that for each y∈[0,1]y \in [0, 1] the equation f(x)=yf(x) = y has uncountably many solutions xx.
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