On continuity of functions with generalized derivatives (Q1735124)

Let \(B\subset\mathbb R^n\) be a ball and \(L_{p,q}(B)\), \(1\leq p,q<\infty,\) be the corresponding Lorentz space. It is known that, if \(u \in W^{1}_{1,\mathrm{loc}}(B)\) and \(\nabla u\in L_{n,1}(B) \), then \(u\) is \(n\)-absolutely continuous [\textit{J. Kauhanen} et al., Manuscr. Math. 100, No. 1, 87--101 (1999; Zbl 0976.26004)]. In the present paper, when \(q>1\) and \(B=B(0,1),\) the author constructs a function \(u\in W_1^1(B)\) with \(\nabla u\in L_{n, q}\) such that, in any neighborhood of \(x=0\), the function \(u\) has an unremovable discontinuity at this point.