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  "body": "\n## Definition\n\nLet $f$ be a function from the class of [[ordinals|0RDJK7EQYY]] to the class of ordinals.\nThe function $f$ is normal if it satisfies both conditions:\n\n1. $f$ is strictly increasing: $\\alpha \u003c \\beta$ implies $f(\\alpha) \u003c f(\\beta)$.\n2. $f$ is continuous: for each nonzero limit ordinal $\\lambda$, $f(\\lambda) = \\sup \\set{f(\\beta) : \\beta \u003c \\lambda}$, the [[supremum|0BTMQ04556]] of the earlier values.\n\n## Remarks\n\nExample: with [[ordinal arithmetic|2C7MBG93AZ]], $f(\\alpha) = \\omega + \\alpha$ is normal.\nIt is strictly increasing, and $\\omega + \\lambda = \\sup_{\\beta \u003c \\lambda} (\\omega + \\beta)$ holds by the definition of addition at limits.\nNon-example: $g(\\alpha) = \\alpha + 1$ is strictly increasing but not continuous.\nIndeed $\\sup_{n \u003c \\omega} g(n) = \\omega$, while $g(\\omega) = \\omega + 1$.\nOther normal functions are $\\alpha \\mapsto \\omega^\\alpha$ and $\\alpha \\mapsto \\aleph_\\alpha$.\nA strictly increasing $f$ always satisfies $f(\\alpha) \\ge \\alpha$.\nFixed-point lemma: for each normal $f$ and each ordinal $\\gamma$, there is an ordinal $\\alpha \\ge \\gamma$ with $f(\\alpha) = \\alpha$.\nThe fixed points form a closed unbounded class, and the function that lists them in increasing order is again normal.\nThe same definition is used for functions from a limit ordinal to itself, or from a regular cardinal to itself.\n",
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