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Riemann equivalence of functions
Published online by Cambridge University Press: 26 February 2010
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Two finite real functions ƒ(x) and g(x), defined for — ∞ < x < ∞, are said to be Riemann equivalent if |ƒ(x)—g(x)| has a zero Riemann integral over every finite interval; we then write ƒ~g or
N. G. de Bruijn conjectured that if ƒ(x+h)~ƒ(x) for every real number h, then ƒ~c where c is a constant; this was proved by P. Erdös [1]. In this note we associate with an arbitrary function ƒ the additive group 
(ƒ) of all numbers h which make ƒ(x)~ƒ(x+h), i.e. which make
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- Copyright © University College London 1955
References
1.Erdös, P., “A theorem on the Riemann integral”, Indag. Math., 14, No. 2 (1952), 142–144.CrossRefGoogle Scholar
2.Souslin, M., “Sur un corps non-dénombrable de nombres réels”, Fundamenta Math., 4 (1923), 311–315.CrossRefGoogle Scholar
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