On functions having Cauchy differences in some prescribed sets

Assume E is a real topological vector space, F is a real Banach space, K is a discrete subgroup of F and C is a symmetric, convex and compact subset of F such that K∩(6C) = {0}. If a function h: E → F is continuous at at least one point and h(x+y)- h(x) - h(y) ∈ K + C for all x, y ∈ E, then there exists a continuous linear function a: E → F such that h(x) - a(x) ∈ K + C for every x ∈ E.