Functions Which Are Symmetric about Several Points
نویسندگان
چکیده
If 1(t) is odd about several points (xe , I (x,)) it is to be understood that the exceptional set may depend on x a . BOAS 1) proves among others that if 1(t) is periodic, bounded on a set of positive measure, and satisfies (1) for a set of x's having positive measure then 1(t) is equivalent to a constant (i .e . 1(t) is constant almost everywhere) . He also shows there exists a bounded periodic function not equivalent to a constant which is odd about a denumerable set of points . He proposes the question if a bounded periodic 1(h exists not equivalent to a constant which is odd about a noncountable set of points . He remarks that it is clearly necessary to put on 1(t) some restriction like boundedness since every additive function (that is, every solution of the functional equation f(x + y) = f(x) + Ay)) satisfies (1) for all x . We shall prove that such functions do exist . We shall also consider the more general functional equation
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