On the essential continuity of summable functions

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 $1<p<\infty$, and let the modulus of continuity $\omega(\delta)$ be such that the series $\sum_{n=1}^\infty n^{1/p-1}\omega(1/n)$ ($1<p<\infty$) 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.