## Real Analysis Exchange

### Darboux symmetrically continuous functions.

Harvey Rosen

#### Abstract

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

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

Dates
First available in Project Euclid: 20 July 2007

https://projecteuclid.org/euclid.rae/1184963809

Mathematical Reviews number (MathSciNet)
MR2010329

Zentralblatt MATH identifier
1046.26002

#### Citation

Rosen, Harvey. Darboux symmetrically continuous functions. Real Anal. Exchange 28 (2002), no. 2, 471--476. https://projecteuclid.org/euclid.rae/1184963809

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