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) = 2 c , 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.
We compare some of the restriction properties that can be found throughout the literature. For example, theorem 10 is a common generalization of three theorems: Blumberg's theorem [2], Baldwin's strengthening of Blumberg's theorem [1], and a related Brown-Prikry's result [8] on Marczewski's (s)-measurable functions.
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