k-to-1 functions on (0,1) (Q1105713)

A function is said to be a k-to-one function if each value it gives is an image of exactly k elements of the domain. The author shows that, for every natural number k, a k-to-one function on (0,1) onto a circle must have infinitely many discontinuities, that if k is \(>2\) the ``figure eight'' is a k-to-one continuous image of either (0,1) or [0,1], and that for each odd \(k>2\) there is a continuous k-to-one function from (0,1) onto itself, but if k is even there must be at least one point of discontinuity.