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$ k$-to-$ 1$ functions on an arc


Authors: Hidefumi Katsuura and Kenneth R. Kellum
Journal: Proc. Amer. Math. Soc. 101 (1987), 629-633
MSC: Primary 54C10; Secondary 54C30, 54F15
DOI: https://doi.org/10.1090/S0002-9939-1987-0911022-X
MathSciNet review: 911022
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Abstract: Recently Jo W. Heath [6] has shown that any $ 2$-to-$ 1$ function from an arc onto a Hausdorff space must have infinitely many discontinuities. Here we investigate extending Heath's result to $ k$-to-$ 1$ functions for $ k > 2$. Examples show that in general Heath's theorem cannot be extended even for functions from an arc into itself. However, if $ f$ is a $ k$-to-$ 1$ function $ (k \geq 2)$ from an arc onto an arc, then we prove that $ f$ has infinitely many discontinuities.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0911022-X
Keywords: $ k$-to-$ 1$ function, $ 2$-to-$ 1$ function
Article copyright: © Copyright 1987 American Mathematical Society

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