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cross the axis uncountably often

Source: IMC 1997 day 2 problem 6

October 22, 2005
functiontrigonometryreal analysisreal analysis unsolved

Problem Statement

Let f:[0,1]→Rf: [0,1]\rightarrow \mathbb{R} continuous. We say that ff crosses the axis at xx if f(x)=0f(x)=0 but ∃y,z∈[x−ϵ,x+ϵ]:f(y)<0<f(z)\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z) for any ϵ\epsilon. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?
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