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Find out if there exists such an function

Source: MTRP 2017 Class 11-Short Answer Type Question: Problem 3 :-

May 26, 2020
functionalgebra

Problem Statement

Let f:[0,1]→[0,1]f:[0,1]\to [0,1] be a continuous function. We say f≡0f\equiv 0 if f(x)=0f(x)=0 for all x∈[0,1]x\in [0,1] and similarly f≢0f\not\equiv 0 if there exists at least one x∈[0,1]x\in [0,1] such that f(x)≠0f(x)\neq 0. Suppose f≢0f\not\equiv 0, f∘f≢0f \circ f \not\equiv 0 but f∘f∘f≡0f \circ f \circ f \equiv 0. Do there exists such an ff? If yes construct such an function, if no prove it
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