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Continuous function with arbitrarily many intersections

Source: ICMC 2023 Round 1 P5

November 28, 2022
college contestsreal analysiscontinuous functionfunctionICMC

Problem Statement

Let [0,1][0, 1] be the set {x∈R:0≤x≤1}\{x \in \mathbb{R} : 0 \leq x \leq 1\}. Does there exist a continuous function g:[0,1]→[0,1]g : [0, 1] \to [0, 1] such that no line intersects the graph of gg infinitely many times, but for any positive integer nn there is a line intersecting gg more than nn times?
Proposed by Ethan Tan
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