Darboux symmetrically continuous functions.

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 \(E\).