On the Continuity of Additive—Like Functions and Jensen Convex Functions which are Borel on a Sphere

Abstract

Assume X is a separable Banach space with dim X ≥ 2.

• 1. If T is a topological space with a countable base, F : T → T is a Borel function such that F(·, t) is continuous for every t ∈ T, f : X → T satisfies

  EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaamiEaiabgUcaRiaadMhaaiaawIcacaGLPaaacqGH9aqpcaWG % gbWaaeWaaeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaai % ilaiaadAgadaqadaqaaiaadMhaaiaawIcacaGLPaaaaiaawIcacaGL % Paaaaaa!4632!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f\left( {x + y} \right) = F\left( {f\left( x \right),f\left( y \right)} \right)$$

  for all x, y ∈ X, and the restriction of f to a sphere is a Borel function, then f is continuous.

• 2. If D is an open and convex subset of X, f : D → ℝ is Jensen convex and the restriction of f to a sphere contained in D is a Borel function, then f is continuous.