Real Analysis Exchange

On Scottish Book Problem 157

Kevin Beanland, Paul D. Humke, and Trevor Richards

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This paper descibes 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 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.

Article information

Real Anal. Exchange, Volume 41, Number 2 (2016), 331-346.

First available in Project Euclid: 30 March 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Secondary: 26A48: Monotonic functions, generalizations

Scottish book 157 approximate derivate approximate continuity monotone


Beanland, Kevin; Humke, Paul D.; Richards, Trevor. On Scottish Book Problem 157. Real Anal. Exchange 41 (2016), no. 2, 331--346.

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