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Characterization of integrands whose primitives can be found by substitution

Source: Romania National Olympiad 2014, Grade XII, Problem 2

March 3, 2019
functionreal analysis

Problem Statement

Let I,J I,J be two intervals, φ:J⟶R \varphi :J\longrightarrow\mathbb{R} be a continuous function whose image doesn't contain 0, 0, and f,g:I⟶J f,g:I\longrightarrow J be two differentiable functions such that f′=φ∘f,g′=φ∘g f'=\varphi\circ f,g'=\varphi\circ g and such that the image of f−g f-g contains 0. 0. Show that f f and g g are the same function.
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