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Claim. In P. Erdős, Problems and results on the theory of interpolation. I, Acta Math. Acad. Sci. Hungar. 9 (1958), no. 3-4, 381-388, printed p. 384, Erdős asserts that a node system in $[-1,1]$ can be built so that, for every continuous $f$, there are continuum many points $x_0$ at which the Lebesgue function is unbounded, his condition (4) on p. 382, that $\limsup_n\sum_{k\le n}\lvert l_k(x_0)\rvert=\infty$, and nevertheless $L_n(f,x_0)\to f(x_0)$. The paper gives no construction. This is a yes to the first question of Problem 671, in a stronger form: the first question asks for one such point for every $f$, and the assertion gives continuum many.

Covers. The first question only. On the same page Erdős writes that when (4) holds at every point of $[-1,1]$ he cannot decide whether some continuous $f$ has $L_n(f,x)$ divergent at every $x$, which is the setting of the second question; the assertion says nothing about it.

Depends on. Nothing in this wiki.

Standing. Withdrawn. P. Erdős and P. Vértesi, On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary system of nodes, Acta Math. Acad. Sci. Hungar. 36 (1980), 71-89, section 1, p. 71, recall the 1958 statement and write that it is perhaps true, that they cannot prove it, and that the original proof was probably incomplete; the corpus's card for the paper's 1981 correction, Erdős and Vértesi 1981, records that sentence. No proof of the 1958 assertion was published. The page is dated by the issue's publication month, September 1958, as its first day. The pending claims that answer both questions yes are on Price's page and QuietMethod's page; the problem's standing derives from those pages alone.