Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we give
characterizations
of the existence of so-called global and local error bounds, for lower
semicontinuous functions defined on complete metric spaces. We thus
provide a
systematic and synthetic approach to the subject, emphasizing the special
case
of convex functions defined on arbitrary Banach spaces (refining the
abstract part
of Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularity
of closed-graph multifunctions between complete metric spaces.