There is no exactly 𝑘-to-1 function from any continuum onto [0,1], or any dendrite, with only finitely many discontinuities

Abstract

Katsuura and Kellum recently proved [8] that any (exactly) k k -to 1 1 function from [ 0 , 1 ] [0,\,1] onto [ 0 , 1 ] [0,\,1] must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly) k k -to- 1 1 function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.