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Lhc i^@*i?|*)c *4

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We define symmetric continuity for functions defined on arbitrary subsets of R. The main result is that when a symmetrically continuous function is defined on a measurable set (a set with the Baire property), then it is continuous almost everywhere (on a residual set, respectively). This generalizes the known result for functions defined on the whole real line.

We construct a symmetrically continuous function f : R → R such that for some X ⊂ R of cardinality continuum f |X is of Sierpiński-Zygmund type. In particular such an f is not countably continuous. This gives an answer to a question of Lee Larson.