An elementary solution to the mountain climbers' problem (Q1291177)

The following result is known as the solution of the mountain climbers' problem. Let \(f_1, f_2\) be locally non-constant continuous functions mapping the compact unit interval \(I\) onto itself. Then there are functions \(g_1,g_2\) with the same properties such that \(f_1\circ g_1=f_2\circ g_2\). This result has been proved independently many times by many authors: \textit{T. Homma} [Kodai Math. Sem. Reports 1952, 13--16 (1952; Zbl 0046.28703)], \textit{R. Sikorski} and \textit{K. Zarankiewicz} [Fundam. Math. 41, 339--344 (1955; Zbl 0064.05501)], \textit{R. Sikorski} [Fundam. Math. 41, 345--350 (1955; Zbl 0064.05502)], \textit{J. S. Lipiński} [Bull. Acad. Pol. Sci., Cl. III 5, 1019--1021 (1957; Zbl 0078.04803)], \textit{J. Mioduszewski} [Colloq. Math. 9, 233--240 (1962; Zbl 0107.27603)], \textit{J. V. Whittaker} [Can. J. Math. 18, 873--882 (1966; Zbl 0144.18006)], \textit{J. P. Huneke} [Trans. Am. Math. Soc. 139, 383--391 (1969; Zbl 0175.34503)], \textit{M. A. McKiernan} [Aequationes Math. 28, 132--134 (1985; Zbl 0561.39005)], \textit{J. E. Goodman, J. Pach} and \textit{C. K. Yap} [Am. Math. Mon. 96, No. 6, 494--510 (1989; Zbl 0674.05018)], \textit{T. Keleti} [Proc. Am. Math. Soc. 117, No. 1, 89--97 (1993; Zbl 0777.26011)], \textit{B. B. Baird} and \textit{K. D. Magill jun.}, Semigroup Forum 55, No. 3, 267--293 (1997; Zbl 0889.20034)]. The present paper differs from the above quoted ones in the argument which is quite elementary, and self-contained.