Normal function on an ordinal

A function $f:\lambda \to \lambda$ on an ordinal is normal when it is strictly increasing and continuous at limit ordinals: $f(\delta )={\sup }_{\alpha <\delta }f(\alpha )$ for every limit $\delta <\lambda$.

For a strictly increasing ordinal class function continuous at limits, start with ${\alpha }_0=0$ and iterate ${\alpha }_{n+1}=f({\alpha }_n)$. If the sequence stabilizes, its stable value is a fixed point. Otherwise its supremum $\alpha$ is a limit and continuity gives $f(\alpha )={\sup }_{n}f({\alpha }_n)={\sup }_n{\alpha }_{n+1}=\alpha$. Starting the iteration above any prescribed ordinal gives arbitrarily large fixed points.