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# fixed points of normal functions

If $f\colon M\rightarrow\mathbf{On}$ is a function from any set of ordinals to the class of ordinals then $\operatorname{Fix}(f)=\{x\in M\mid f(x)=x\}$ is the set of fixed points of $f$. $f^{\prime}$, the *derivative* of $f$, is the enumerating function of $\operatorname{Fix}(f)$.

If $f$ is $\kappa$-normal then $\operatorname{Fix}(f)$ is $\kappa$-closed and $\kappa$-normal, and therefore $f^{\prime}$ is also $\kappa$-normal.

For example, the function which takes an ordinal $\alpha$ to the ordinal $1+\alpha$ has a fixed point at every ordinal $\geq\omega$, so $f^{\prime}(\alpha)=\omega+\alpha$.

Defines:

derivative

Related:

ProofOfPowerRule, LeibnizNotation, ProofOfProductRule, ProofOfSumRule, SumRule, DirectionalDerivative, NewtonsMethod

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03E10*no label found*

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