Problem:
Discontinuous Counter example
If f:[a,b] -> R is ONE-TO-ONE and satisfies the intermediate value property, then f is continuous on [a,b].
I know that this is a false statement if you exclude the one-to-one property. The example I received before was f(x) = sin(1/x), but this function is not one-to-one. I am having a really hard time coming up with an example of a ONE-TO-ONE function that satisfies the IVP and is discontinuous on an interval. Can you please help me?
Also, is there anyway of correcting this statement to make it true, and if so, can you prove it?