Cardinal invariants concerning extendable and peripherally continuous functions


Krzysztof Ciesielski & Ireneusz Reclaw

Real Anal. Exchange 21 (1995-96), 459-472.

Let G be a family of real functions f:R-->R. In the paper we examine the following question: for which families F of real functions does there exist g:R-->R such that f+g belongs to G for all f from F? More precisely, we will study a cardinal function add(G) defined as the smallest cardinality of a family F of real functions for which there is no such g. We prove that add(ext)=add(pr)=c+ and add(pc)=2c, where c denotes the cardinality of the continuum, and 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, equation add(ext)=c+ implies immediately that every real function is a sum of two extendable functions. This solves a problem of Gibson.

We also study the multiplicative analogue mul(G) of the function add(F) and prove that mul(ext)=mul(pr)=2 and add(pc)=c.

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