Regressive function ($f(\alpha )<\alpha$)

A function on a set of ordinals with $f(\alpha )<\alpha$ at every nonzero point of its domain. On a stationary subset of a regular uncountable cardinal, Fodor lemma makes it constant on a stationary subset.

A regressive function on a stationary subset of a regular uncountable cardinal is constant on a stationary subset.

If $S\subseteq \kappa$ is stationary and $f(\alpha )\in A_{\alpha }$ for a kappa-filtration, then $f$ is constant on a stationary subset. Restrict to limit indices and use continuity to find a smaller stage containing each value. Fodor lemma fixes that stage on a stationary subset. Its size is less than $\kappa$, so club filter completeness makes one value fiber stationary.