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Abstract. Let E be a compact subset of the complex plane cg such that Leja's extremal function L~ for E is continuous. If almost all zeros of the polynomials of best approximation to a function f~C(E) are outside the set E R = {z~Cg:LE(z)< R}, for some R> 1, then f is extendible to a holomorphic function in E R. If the zeros of n-th polynomial of best approximation to fare outside Eg. and the sequence {R, -n} rapidly decreases to zero thenfcan be extended to a C ~ function on ~2. In this paper Bernstein's theorem is proved in the case of uniform approximation on a compact subset E of cg such that Leja's extremal function Le associated with E is continuous (see Corollary 4). A similar result for the polynomials of best approximation to C ~ functions on sufficiently regular compact subsets of cg is also proved (Theorem 9).Let E be a compact subset of the complex plane. Denote by Eoo the unbounded connected component of cg \ E. Throughout the paper we assume that E is not thin at any point of the boundary of Eo~ (or: each

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