Sum of graphs of continuous functions and boundedness of additive operators

Abstract

Assume that f : D 1 → R and g : D 2 → R are uniformly continuous functions, where D 1 , D 2 ⊂ 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 .