It's counter intuitive because it's obviously false as stated: f(x) = 1 + x for x in R. They left out the main assumption of the theorem, namely that the domain

It's counter intuitive because it's obviously false as stated: f(x) = 1 + x for x in R. They left out the main assumption of the theorem, namely that the domain of the function is compact, and gets mapped (in)to itself.