On functions having Cauchy differences in some prescribed sets

AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK ∩ (6C) = {0}. If a functionh:E → F is continuous at at least one point andh(x + y) − h(x) − h(y) ∈ K + C for allx, y ∈ E, then there exists a continuous linear functiona:E → F such thath(x) − a(x) ∈ K + C for everyx ∈ E.