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Katsuura and Kellum recently proved [8] that any (exactly) k-to1 function from [0,1] onto [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-to-1 function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the afflrmative.

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