Symmetrically continuous functions on various subsets of the real line. (English) Zbl 1016.26003

The author studies basic properties of symmetrically continuous functions defined on arbitrary subsets of the real line \(\mathbb{R}\). Let \(A\subset\mathbb{R}\). A function \(f: A\to\mathbb{R}\) is said to be symmetrically continuous at a point \(x\) if for every \(\varepsilon> 0\) there exists \(\delta> 0\) such that for each \(h\), \(|h|<\delta\), \(|f(x+ h)- f(x- h)|<\varepsilon\) holds whenever \(x+ h\in A\) and \(x- h\in A\). The function \(f\) is said to be symmetrically continuous on \(A\) if it is symmetrically continuous at every point of \(A\). The main result of the paper: Let \(f: A\to\mathbb{R}\), where \(A\) is an arbitrary subset of \(\mathbb{R}\). Then the set of points at which \(f\) is symmetrically continuous but not continuous has the inner measure zero and contains no second category set having the Baire property. This statement has an immediate corollary: Let \(E\) be measurable (have the Baire property) and \(f: E\to\mathbb{R}\) be symmetrically continuous. Then \(f\) is continuous almost everywhere on \(E\) (at every point of \(E\) except on a meager set). The above results in the case \(A= E=\mathbb{R}\) were formulated by C. L. Belna [Proc. Am. Math. Soc. 87, 99-102 (2000; Zbl 0515.26003)].


26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable


Zbl 0515.26003