  We study the convergence of Bernstein type operators leading to two results.
The first: The kernel $K_n$ of the Bernstein-Durrmeyer operator at each point
$x \in (0, 1)$ $\unicode{x2013}$ that is $K_n(x, t) dt$ $\unicode{x2013}$ once
standardised converges to the normal distribution. The second result computes
the pointwise limit of a generalised Bernstein-Durrmeyer operator applied to
$\unicode{x2013}$ possibly discontinuous $\unicode{x2013}$ functions $f$ of
bounded variation.
