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Eberly College of Arts and Sciences




Let F be a family of real functions, F ⊆ R R . In the paper we will examine the following question. For which families F ⊆ R R does there exist g : R → R such that f + g ∈ F for all f ∈ F? More precisely, we will study a cardinal function A(F) defined as the smallest cardinality of a family F ⊆ R R for which there is no such g. We will prove that A(Ext) = A(PR) = c + and A(PC) = 2c , where Ext, PR and PC stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from R into R, respectively. In particular, the equation A(Ext) = c + immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson [6]. We will also study the multiplicative analogue M(F) of the function A(F) and we prove that M(Ext) = M(PR) = 2 and A(PC) = c. This article is a continuation of papers [10, 3, 12] in which functions A(F) and M(F) has been studied for the classes of almost continuous, connectivity and Darboux functions.

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