1. The problem statement, all variables and given/known data Let f : (-1, 1) → ℝ. f satisfies the intermediate value property and is one-to-one on (-1, 1). Prove f is continuous on (-1, 1) 3. The attempt at a solution I was thinking that the IVP and one-to-one implies that f should be strictly monotonic and that a strictly monotonic one-to-one function is continuous. Both of these seem very intuitive to me and yet I have no idea how to do the rigorous proof. For example if f is not strictly monotonic then it seems there would be a contradiction in f one-to-one but does that follow directly? Since you don't know anything else about the function..