RESUMEN

We study two loosely related problems concerning approximation propertiesof Bernstein polynomials Bnf(x) of some function f on [0, 1]: theabsence of Gibbs phenomenon at points at which f has jumps, and theconvergence of (Bnf)0(x) towards f0(x). Both results are obtained usingclassical probabilistic tools. In particular the proof of the secondstatement relies on the representation of the derivative (Bnf)0(x) as the expectation of the functional of a random variable.