We improve the degree of pointwise approximation of continuous functions f (x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x) = x log(x) + (1 − x) log(1 − x). At the end we prove a general statement for pointwise approximation by Bernstein operators.2010 Mathematics subject classification: primary 41A10; secondary 41A15, 41A25, 41A36.