Let X be a metric spaces. Let Y be a Banach space partially ordered by a

pointed closed convex cone K. Let Φ be a separable linear family (a class) of Lipschitz

functions defined on X and with values in Y . Let Ω be an open subset of X. We say

that a multifunction Γ mapping Ω into Y is monotone if for all φx ∈ Γ(x),φy ∈ Γ(y)

we have

φx(x) + φy(y) − φx(y) − φy(x) ≥K 0.

In the paper it is proved that under certain conditions on Φ each monotone multifunction

is single-valued and continuous on a dense Gδ-set.