On Scottish Book Problem 157

TL;DR

This paper investigates the solution to Scottish Book Problem 157, revealing that existing theorems and references point to solutions from Ornstein and O'Malley, and discusses properties of related functions.

Contribution

It identifies the first solution to SB 157 through analysis of historical references and unifies the methods used by Ornstein and O'Malley.

Findings

O'Malley's 1975 theorem provides an immediate solution.

Ornstein's 1970 paper likely contains the first solution.

Functions similar to Ornstein's example are typical in the Baire category sense.

Abstract

This paper describes our hunt for the solver of Problem 157 in the Scottish Book, a problem originally posed by A.~J. (Gus) Ward in 1937. We first make the observation that a theorem of Richard O'Malley from 1975 yields an immediate positive solution. A further look at O'Malley's references revealed a 1970 paper by Donald Ornstein that we now believe contains the first solution of {\em SB 157}. We isolate the common elements in the machinery used by both Ornstein and O'Malley and discuss several consequences. We also examine an example function given by Ornstein. There are some difficulties with this function but we provide a fix, and show moreover that functions of that kind are typical in the sense of the Baire category theorem.