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2002/2003 Darboux symmetrically continuous functions.
Harvey Rosen
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Real Anal. Exchange 28(2): 471-476 (2002/2003).


For a symmetrically continuous function \(f : \mathbb R\to [0,1]\), a reduction formula is obtained which gives a Darboux symmetrically continuous function \(g_f : \mathbb R\to [0,1]\) such that the set \(C(f)\) of continuity points of \(f\) is a subset of \(C(g_f)\). Under additional conditions, \(g_f\) and the oscillation function \(\omega_f\) of \(f\) are Croft-like functions. One consequence of \(g_f\) being Darboux is that the absolutely convergent values \(s(x)\) of a real trigonometric series \(\sum_{n=1}^{\infty} \rho_n \sin(nx + x_n)\), with \(\sum_{n=1}^{\infty} | \rho_n |=\infty\) and with an uncountable set \(E\) of points of absolute convergence, almost has the intermediate value property except for countably many values \(s(x)\) and countably many points of \(E\).


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Harvey Rosen. "Darboux symmetrically continuous functions.." Real Anal. Exchange 28 (2) 471 - 476, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1046.26002
MathSciNet: MR2010329

Primary: 26A15

Keywords: Darboux function , symmetric continuity , trigonometric series

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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