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MR927692 (89b:54017) 54C10 (54F15 54F50)
Heath, Jo W. There is no exactly $k$-to-$1$ function from any continuum onto $[0,1]$, or any dendrite, with only finitely many discontinuities. Trans. Amer. Math. Soc. 306 (1988), no. 1, 293–305.
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