ABSTRACT.

It is shown that  the m-th order derivative of the
n-th order Bernstein polynomial of a function f satisfying
a certain Lipschitz condition, can be written for n diverging to infinity  
as  a  singular integral  of Gauss-Weierstrass type, m times
differentiated (in a certain sense) under the integral sign.

The theorem is applied to yield a formula 
involving p times differentiated Bernstein polynomials of functions that are not 
p times continuously differentiable.