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

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