Abstract
This paper studies the relation between the integral smoothness of a function and its essential continuity, and also the convergence of Steklov means and Fourier series. Let , and let the modulus of continuity be such that the series diverges. Then in the class there is a bounded function with the following properties: 1) cannot be altered on a set of measure zero so as to obtain a function continuous at even one point. 2) If is an arbitrary positive sequence with , then there is a set of second category such that the sequence diverges at each point . 3) The partial sums of the Fourier series of are uniformly bounded. 4) For any sequence , , there is a set of second category such that diverges for each . Bibliography: 16 titles.
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