A new, general criterion is given for ensuring that a closed saddle function has a nonempty compact set of saddlepoints. Under this criterion it is shown also that every minimaximizing sequence clusters around some saddlepoint. A comparable theorem is given for semicontinuous quasi-saddle functions. The new criterion is applied to constrained saddlepoint problems and to the Fenchel-Rockafellar duality model for constrained convex minimization. Finally, the relationship to existing saddlepoint results is explored in detail.