𝐾-to-1 functions on arcs for 𝐾 even

Heath, Jo W.

doi:10.1090/S0002-9939-1987-0902560-4

$K$-to-$1$ functions on arcs for $K$ even
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Proc. Amer. Math. Soc. 101 (1987), 387-391 Request permission

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.

References

Paul W. Gilbert, $n$-to-one mappings of linear graphs, Duke Math. J. 9 (1942), 475–486. MR 7106
O. G. Harrold Jr., Exactly $(k,1)$ transformations on connected linear graphs, Amer. J. Math. 62 (1940), 823–834. MR 2554, DOI 10.2307/2371492
Jo Heath, Every exactly $2$-to-$1$ function on the reals has an infinite set of discontinuities, Proc. Amer. Math. Soc. 98 (1986), no. 2, 369–373. MR 854049, DOI 10.1090/S0002-9939-1986-0854049-8

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functions on an arc