Sorry, I forgot you included the qualification about the derivative being zero at only countably many points.
It's not clear to me why, with that qualification, we have now a necessary and sufficient condition for a differentiable function on the line being injective.
I can certainly...
No, that still cannot be correct, and for the same reason, namely, a constant function (and even a function that's nondecreasing and constant on some interval) would satisfy the condition yet not be injective.
Statement (1) is wrong because a constant function would satisfy the condition yet not be injective! Certainly a strictly positive derivative everywhere or a strictly negative derivative everywhere suffices to be injective. As already mentioned, there is an issue about what kind of zeros the...
I innocently gave my students a problem: Which differentiable functions f: R \rightarrow R are bijective? "Innocently", I say, because I'm finding it hard to come up with any simple set of conditions that are both necessary and sufficient. Here's what I can say so far:
(1) If f'(x) \neq 0...