An elementary solution to the mountain climbers' problem

Summary.

In this paper the following well known result is proved in a constructive elementary way: if f ₁, f ₂ : [0, 1] → [0,1] are locally non-constant continuous functions with f ₁ (0) = f ₂ (0) = 0, f ₁ (1) = f ₂ (1) = 1, then there are functions g ₁, g ₂ with the same properties such that $f_1\circ g_1=f_2\circ g_2$.