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We show that under a condition on the Dirichlet integral, Sobolev functions with zero boundary values at the points of a set of positive capacity are identically zero.

1. Main result. All zeros of nonconstant analytic functions of the complex plane are isolated by a basic topological property of analytic functions. Results on the size of the zero set of the radial limits or nontangential boundary values of analytic functions of the unit disk and some related classes of functions have been proved by several authors. We now briefly review such theorems.