This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L∞. In 1993, Bourgain showed that for a real-valued bounded function f, the set Ef of B-points is everywhere dense and has the maximal Hausdorff dimension, dim H(Ef ) = 1; in addition, the vertical variation of the harmotic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles. © 2009 Springer Science+Business Media, Inc.