  Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the
measure formed by adding a pure point to $d\mu$. We give a simple formula for
the Verblunsky coefficients of $d\nu$ based on a result of Simon.
  Then we consider $d\mu_0$, a probability measure on the unit circle with
$\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of
bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the
probability measure $d\mu_m$. We use the formula above to prove that the
Verblunsky coefficients of $d\mu_m$ are in the form $\alpha_n(d\mu_0) +
\sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n$, where the $c_j$'s are constants
of norm 1 independent of the weights of the pure points and independent of $n$;
the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that
$d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall
introduce in the paper. Then we use this fact to prove that $\lim_{n \to
\infty} \vp_n^*(z, d\mu_m)$ is continuous and is equal to $D(z, d\mu_m)^{-1}$
away from the pure points.
