The mountain climbers’ problem

Abstract:

We show that two climbers can climb a mountain in such a way that at each moment they are at the same height above the sea level, supposing that the mountain has no plateau. That is, if $f$ and $g$ are continuous functions mapping $[0,1]$ to $[0,1]$ with $f(0) = g(0) = 0$ and $f(1) = g(1) = 1$, and if neither $f$ nor $g$ has an interval of constancy then there exist continuous functions $k$ and $h:[0,1] \to [0,1]$ satisfying $k(0) = h(0) = 0,\;k(1) = h(1) = 1$, and $f \circ k = g \circ h$.