In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended real number line In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to It is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to at a point specifically to that point from even being considered as a potential solution (to the minimization problem). Points at which the function takes the value (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem, with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to at that point instead. When a minimum point (in ) of a function is to be found but 's domain is a proper subset of some vector space then it often technically useful to extend to all of by setting at every By definition, no point of belongs to the effective domain of which is consistent with the desire to find a minimum point of the original function rather than of the newly defined extension to all of If the problem is instead a maximization problem (which would be clearly indicated) then the effective domain instead consists of all points in the function's domain at which it is not equal to Definition Suppose is a map valued in the extended real number line whose domain, which is denoted by is (where will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the of is denoted by and typically defined to be the set unless is a concave function or the maximum (rather than the minimum) of is being sought, in which case the of is instead the set In convex analysis and variational analysis, is usually assumed to be unless clearly indicated otherwise. Characterizations Let denote the canonical projection onto which