In this paper, we generalize some results of Stein and Zygmund and of Evans and Larson concerning symmetric functions. In particular, we show that if f is Lebesgue measurable or has the Baire property in the wide sense, then the set of symmetric points of / is Lebesgue measurable or has the Baire property in the wide sense, respectively. We also give some examples that show that these results cannot be improved in a certain sense. Finally, we show that there are plenty of examples of functions

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... hat are both Lebesgue measurable and have the Baire property in the wide sense, yet the set of points where each of the functions is symmetric and discontinuous has the same cardinality as that of the continuum.