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 .

Approximate Homomorphisms and Derivations on Normed Lie Triple Systems

“…So ( + ) = ( ) + ( ) for all ∈ , and then is an additive mapping and ( ) = ( ) for all ∈ and all ∈ . Hence the assumptions of Lemma 1 in [31] are fulfilled, for instance, in view of the comments in the papers [32,33], and so we can deduce the lemma. The converse is trivial.…”

“…This and (26) show that is additive. Consequently, (26) and Lemma 1 in [31] jointly with the remarks in the papers [32,33] imply the Clinearity of . Further, it follows from (5) and (8) that, for all , , ∈ and all ∈ N,…”

Survey on Recent Ulam Stability Results Concerning Derivations

“…For instance, if a set ⊂ F (with F ∈ {R, C}) has a positive inner Lebesgue measure or contains a subset of the second category and with the Baire property, then int( − ) ̸ = 0 (see, e.g., [56]). For related results we refer to [56][57][58].…”

“…[56]). For more information on B F and further references concerning the subject, we refer the reader to, for example, [56][57][58].…”