  We prove an analytic version of the stable graph regularity lemma from
\cite{MaSh}, which applies to stable functions $f\colon V\times W\to [0,1]$.
Our methods involve continuous model theory and, in particular, results on the
structure of local Keisler measures for stable continuous formulas. Along the
way, we develop some basic tools around ultraproducts of metric structures and
linear functionals on continuous formulas, and we also describe several
concrete families of examples of stable functions.
