Normal function

In ordinal analysis, a normal function is a function \(f: \text{On} \mapsto \text{On}\) that is strictly increasing and continuous. That is, \(\alpha < \beta \Rightarrow f(\alpha) < f(\beta)\) and \(\alpha \in \text{Lim} \Rightarrow f(\alpha) = \sup\{\beta : \beta < \alpha\}\).

Normal functions are subject to an important property given by the fixed-point lemma, which states that a normal function has arbitrarily large fixed points.