A note on a theorem of continuum of zero points

By developing an algorithm, Herings, Talman and Yang recently proved the following interesting and deep theorem: A correspondence ζ from the n-dimensional unit cube Uⁿ to n-dimensional Euclidean space Rⁿ has a continuum of zero points containing the origin and the vector of all-ones if the correspondence satisfies certain conditions. In this note we give an alternative proof of the theorem in the case where the correspondence ζ : Uⁿ → Rⁿ is single-valued.