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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ K$-to-$ 1$ functions on arcs for $ K$ even


Author: Jo W. Heath
Journal: Proc. Amer. Math. Soc. 101 (1987), 387-391
MSC: Primary 26A15; Secondary 54C10
MathSciNet review: 902560
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Abstract: For exactly $ k$-to-$ 1$ functions from $ [0,1]$ into $ [0,1]$:

(1) at least one discontinuity is required (Harrold),

(2) if $ k = 2$, then infinitely many discontinuities are needed, for any Hausdorff image space (Heath),

(3) if $ k = 4$, or if $ k$ is odd, then there is such a function with only one discontinuity (Katsuura and Kellum), and, it is shown here that

(4) if $ k$ is even and $ k > 4$, then there is such a function with only two discontinuities, and no such function exists with fewer discontinuities.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0902560-4
PII: S 0002-9939(1987)0902560-4
Keywords: $ k$-to-$ 1$ function, $ k$-to-$ 1$ map
Article copyright: © Copyright 1987 American Mathematical Society