When does the sequence $u_{n+1}=f(u_n)$ converge?

Question

I have two questions:

1) Is there an example of a continuous function $f$ with $f[0,1]\subset [0,1]$ such that the recursive sequence $u_{n+1}=f(u_n)$ with $u_0\in [0,1]$ does not converge?

2) Is there a characterization of continuous functions $f$ with $f[0,1]\subset [0,1]$ such that every recursive sequence $u_{n+1}=f(u_n)$ with $u_0\in [0,1]$ is convergent?

Answer 1

As regards the first question if $f(x)=1-x$ then the recursive sequence does not converge for any $u_0\in [0,1]\setminus\{1/2\}$.

Any continuous function $f$ such that $f[0,1]\subset [0,1]$ has a fixed point. So there is at least one point $u_0$ such that the recursive sequence does converge.

For the second question, any contraction mapping has the required property but I don't know the characterization that you are asking for.