---
tag: PXTF18X433
kind: definition
name: Normal function
msc: ["03E10"]
defines: ERTAM0SFVB
uses: [0RDJK7EQYY, 0BTMQ04556, 2C7MBG93AZ]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_function
---

## Definition

Let $f$ be a function from the class of [[ordinals|0RDJK7EQYY]] to the class of ordinals.
The function $f$ is normal if it satisfies both conditions:

1. $f$ is strictly increasing: $\alpha < \beta$ implies $f(\alpha) < f(\beta)$.
2. $f$ is continuous: for each nonzero limit ordinal $\lambda$, $f(\lambda) = \sup \set{f(\beta) : \beta < \lambda}$, the [[supremum|0BTMQ04556]] of the earlier values.

## Remarks

Example: with [[ordinal arithmetic|2C7MBG93AZ]], $f(\alpha) = \omega + \alpha$ is normal.
It is strictly increasing, and $\omega + \lambda = \sup_{\beta < \lambda} (\omega + \beta)$ holds by the definition of addition at limits.
Non-example: $g(\alpha) = \alpha + 1$ is strictly increasing but not continuous.
Indeed $\sup_{n < \omega} g(n) = \omega$, while $g(\omega) = \omega + 1$.
Other normal functions are $\alpha \mapsto \omega^\alpha$ and $\alpha \mapsto \aleph_\alpha$.
A strictly increasing $f$ always satisfies $f(\alpha) \ge \alpha$.
Fixed-point lemma: for each normal $f$ and each ordinal $\gamma$, there is an ordinal $\alpha \ge \gamma$ with $f(\alpha) = \alpha$.
The fixed points form a closed unbounded class, and the function that lists them in increasing order is again normal.
The same definition is used for functions from a limit ordinal to itself, or from a regular cardinal to itself.