Question 81
Let $I$, $J$ be intervals and consider $\varphi : J \to \mathbb{R}$ a continuous function which is nonzero on $J$. Let $f$, $g : I \to J$ be two differentiable functions such that $f' = \varphi \circ f$ and $g' = \varphi \circ g$. Prove that if there exists $x_0 \in I$ such that $f(x_0) = g(x_0)$, then $f$ and $g$ coincide.