Some theorems on passing from local to global presence of properties of functions

TL;DR

This paper establishes theorems that connect local properties of functions on specific sets to their global properties, providing conditions for when local restrictions imply a global function within various classes.

Contribution

It introduces new theorems that characterize when local properties of functions guarantee a global function in classes like partial recursive, continuous, and computable functions.

Findings

Necessary and sufficient conditions for strongly join permitting classes of partial recursive functions.

A sufficient and necessary condition for continuous functions between topological spaces.

Extension of the results to computable functions in effective topological spaces.

Abstract

When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words "a subset of"). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some $n$, the class consists of the potentially partial recursive functions of $n$ variables, and the collection consists of sets of $n$-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and the collection consists of subsets of the first of them (the condition is also necessary under a weak assumption on the second one). The third theorem is of a similar character but, instead of continuity, it concerns computability in the spirit of the one in effective topological spaces.