Given x 0 , a point of a convex subset C of a Euclidean space, the two following statements are proven to be equivalent: (i) every convex function f : C → ℝ is upper semi-continuous at x 0 , and (ii) ...

Let 𝓟 be an ideal of closed subsets of a T₁ topological space X. Suppose C(X)𝓟 is the ring of all real-valued functions on X whose closure of discontinuity set is a member of 𝓟. We investigate a ...