I have the 4th edition of Spivak's Calculus. Problem 13(b) in Chapter 7 says: ------ Suppose that f satisfies the conclusion of the Intermediate Value Theorem, and that f takes on each value only once. Prove that f is continuous. ------- Well, what about this function: f(1) = 2 f(2) = 1 f(x) = x for all other x And if you look at this on the interval [0, 3] then certainly for every c between f(0)=0 and f(3)=3 there is an x such that f(x)=c, so the IVT conclusion is satisfied. And f takes on each value only once. But f isn't continuous. So isn't this a counterexample to what I'm supposed to prove?