Every exactly 2-to-1 function on the reals has an infinite set of discontinuities

Heath, Jo

doi:10.1090/S0002-9939-1986-0854049-8

Every exactly $2$-to-$1$ function on the reals has an infinite set of discontinuities
HTML articles powered by AMS MathViewer

by Jo Heath

Proc. Amer. Math. Soc. 98 (1986), 369-373

DOI: https://doi.org/10.1090/S0002-9939-1986-0854049-8

PDF | Request permission

Abstract:

It has long been known that the set of discontinuities of a $2$-to-$1$ function on either the closed or the open interval must be nonempty; this paper proves that the set must be infinite.

References

Karol Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1957), 84–98. MR 102063, DOI 10.4064/fm-45-1-84-98
Paul Civin, Two-to-one mappings of manifolds, Duke Math. J. 10 (1943), 49–57. MR 8697
Paul W. Gilbert, $n$-to-one mappings of linear graphs, Duke Math. J. 9 (1942), 475–486. MR 7106
O. G. Harrold Jr., The non-existence of a certain type of continuous transformation, Duke Math. J. 5 (1939), 789–793. MR 1358
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
Venable Martin and J. H. Roberts, Two-to-one transformations on 2-manifolds, Trans. Amer. Math. Soc. 49 (1941), 1–17. MR 4129, DOI 10.1090/S0002-9947-1941-0004129-9
J. Mioduszewski, On two-to-one continuous functions, Rozprawy Mat. 24 (1961), 43. MR 145490
Sam B. Nadler Jr. and L. E. Ward Jr., Concerning exactly $(n,\,1)$ images of continua, Proc. Amer. Math. Soc. 87 (1983), no. 2, 351–354. MR 681847, DOI 10.1090/S0002-9939-1983-0681847-3
J. H. Roberts, Two-to-one transformations, Duke Math. J. 6 (1940), 256–262. MR 1923