"For a given real, continuous function $f: \mathbb{R} \to \mathbb{R}$ and arbitrarily chosen $x_1$, and define $(x_n)$ by setting $x_{n+1} = f(x)$."
Am I correct in thinking that the orbit of any sequence constructed as above will either tend towards a fixed point or will diverge? There are no real, continuous functions which cause a sequence to oscillate endlessly?