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## Homework Statement

Suppose f: ℝ → ℝ takes on each of its values exactly twice; that is, for each y in ℝ, the set {x: y = f(x)} has either 0 or 2 elements. Show that f is discontinuous at infinitely many points.

## Homework Equations

I don't know if this is relevant, but in the prior text to this problem, this was perhaps the only one that may be relevant:

If I is an interval in ℝ and if f: I → ℝ is a nonconstant continuous function, then f(I) is an interval. In particular, if a,b in I with f(a) ≠f(b), then f assumes every value between f(a) and f(b)

## The Attempt at a Solution

I've tried going through definitions, theorems, etc. for several hours of the course of 3 days and I am still totally stumped. Ugh. Please, any help is much appreciated.