## 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)].

### MSC:

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

### Keywords:

symmetrically continuous functions

Zbl 0515.26003