Real Analysis Exchange

Darboux symmetrically continuous functions.

Harvey Rosen

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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 \(E\).

Article information

Real Anal. Exchange, Volume 28, Number 2 (2002), 471-476.

First available in Project Euclid: 20 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

symmetric continuity Darboux function trigonometric series


Rosen, Harvey. Darboux symmetrically continuous functions. Real Anal. Exchange 28 (2002), no. 2, 471--476.

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