Sum of graphs of continuous functions and boundedness of additive operators

Assume that f :D1→R and g :D2→R are uniformly continuous functions, where D1,D2 ⊂ X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f (x) = x∗(x) + a and g(x) = x∗(x) + b with some x∗ ∈ X∗ and a, b ∈ R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X ×R treated as a normed space with a norm (x,α) = x 2 + |α|2.  2005 Elsevier Inc. All rights reserved.