I will present recent work with N. Chavarria and A. Pillay on an analytic version of the Malliaris-Shelah stable graph regularity lemma, 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. I will also describe how stability of Hilbert spaces (in continuous logic) gives rises to several concrete families of examples of stable functions motivated by various areas of combinatorics.