Mountain climbing: An alternate proof (Q2266393)

The paper deals with the so-called ''Mountain climbing problem'', proposed by \textit{J. V. Whittaker} [Can. J. Math. 18, 873-882 (1966; Zbl 0144.180)] and solved by \textit{J. P. Huneke} [Trans. Am. Math. Soc. 139, 383- 391 (1969; Zbl 0175.345)]. The author presents an alternate proof of the following theorem: Assume that f and g map [0,1] onto [0,1] and are continuous, locally non-constant, with \(f(0)=g(0)=0\), \(f(1)=g(1)=1\). Then there exist functions \(\phi\) and \(\psi\) having these same properties and satisfying \(f\circ \phi =g\circ \psi\) on [0,1]. The new proof is based on the properties of continua in the plane and is shorther than that of Huneke.