Limit of a monotone function

Question

Let $f\colon [a,b] \to[a,b]$ be a non-decreasing function in a sense that $f(x)\leq f(y)$ whenever $x\leq y$. Although there may be several fixpoints of $f$, at least one does always exist and there exists a greatest fixpoint, let us call it $x^*$. I wonder whether it holds that $x^* = \lim\limits_n f^n(b)$. Clearly, if $f$ is continuous this would be true, but I wonder whether this fact can be established purely based on the monotonicity of $f$.

Answer 1

I think the answer is no, and the example is given as follows. Let $a = -1$ and $b = 1$, with $$ f(x) = \begin{cases} \frac12x^2,& \text{ if }x>0 \\ -\frac12,&\text{ if }x\leq 0. \end{cases} $$ We have that $x^* = -\frac12$ whereas $f^n(b) \geq 0$ for all $n $ and also $\lim_n f^n(b) = 0$. Hence, motonocity is not enough by itself and some continuity assumption is needed.