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 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.
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