Convergence of partial maps

Given metric spaces ( X , d ) and ( Y , ρ ) , a partial map between X and Y is a pair ( D , u ) , where D is a closed subset of X and u : D → Y is a function. We introduce a general convergence notion for nets of such partial functions. While our initial description is variational in nature, we show that this description amounts to bornological convergence of the associated net of graphs as defined by Lechicki, Levi and Spakowski [26] with respect to a natural bornology on X × Y , and which places the work on continuous partial functions of Brandi, Ceppitelli, and Holá [12,13,20,21] in a general framework.