Simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition

Let E be a compact subset of the complex plane ?, having positive planar Lebesgue measure. Then there exists a nonconstant function f, analytic in the domain ? É, satisfying the Lipschitz condition In this note there is given a simple proof of the theorem of N. X. Uy, formulated above. It is also proved that each bounded measurable function α, defined on the set E, can be revised on a set of small Lebesgue measure so that for the function ? obtained the Cauchy integral satisfies condition (1).