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For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality f(x′0) + f (x′1) ⩽ f(x0) + f(x1), for x′i ∊ [x0, x1], for x′i ∊[x0, x1], ∥x′i - xi∥ = γ, i = 0, 1, whenever x0, x1 ∊ D and ∥x0 - x1∥ ➮ γ. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x* ∊ D such that f is bounded below on D ∩ B (x*, γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball B(x*, γ/2) ⊂ D and D′ is a closed bounded subset of D, then f is bounded on D′ if it is bounded above on the boundary of D′; (c) if dim X > 1 and the interior of D contains a closed ball of radius γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x*, γ/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D. [ABSTRACT FROM AUTHOR]

Noiri in 1984, Dontchev, Ganster and Reilly in recent years and Reilly and Vamanamurthy in 1983 introduced the notion of perfectly continuous, regular set-connected and clopen functions, respectively. The aim of this paper is to introduce the notion of a new class of functions which is called almost clopen functions including the classes of perfectly continuous, regular set-connected and clopen functions. Furthermore, properties of almost clopen functions are obtained and relationships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity are investigated. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to investigate some properties of H-continuous (Almost Hcontinuous) multifunctions defined from a product space to a product space. The relations between the strongly closed graph of a multivalued functions and H-upper semicontinuity of multivalued functions are also investigated. [ABSTRACT FROM AUTHOR]