In 2011, Malliaris and Shelah proved a strong form of Szemeredi's regularity lemma for the class of "stable graphs", which are graphs omitting a certain special subgraph called a "half-graph". Using tools from additive combinatorics, Terry and Wolf proved a group theoretic analogue of this result in which finite graphs are replaced by subsets of finite abelian groups. A suitable generalization to arbitrary finite groups was then proved by myself, Pillay, and Terry using model theoretic methods. This talk will focus on an analytic analogue of stability defined for bounded functions on groups. Roughly speaking, the main result of the talk says that if G is amenable, then any stable function on G is almost constant on all translates of a unitary Bohr set in G of bounded complexity. I will also discuss several applications related to Bogolyubobv's Lemma, the Croot-Sisask Lemma, and Folner's Theorem.