Geodesic
On the essential continuity of summable functions
Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 301-322 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $1, and let the modulus of continuity $\omega(\delta)$ be such that the series $\sum_{n=1}^\infty n^{1/p-1}\omega(1/n)$ ($1) diverges. Then in the class $H_p^{\omega}$ there is a bounded function $f$ with the following properties: 1) $f$ cannot be altered on a set of measure zero so as to obtain a function continuous at even one point. 2) If $\{h_k\}$ is an arbitrary positive sequence with $h_k\to 0$, then there is a set $E$ of second category such that the sequence $(2h_k)^{-1}\int_{x-h_k}^{x+h_k}f(t)\,dt$ diverges at each point $x\in E$. 3) The partial sums $S_n(f;x)$ of the Fourier series of $f$ are uniformly bounded. 4) For any sequence $\{n_k\}$, $n_k\to\infty$, there is a set $E$ of second category such that $S_{n_k}(f;x)$ diverges for each $x\in E$. Bibliography: 16 titles. 
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V. I. Kolyada. On the essential continuity of summable functions. Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 301-322. http://geodesic.mathdoc.fr/item/SM_1980_36_3_a1/

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