It is known that the ring $B(R)$ of all Baire functions
carrying the pointwise convergence yields a sequential
completion of the ring $C(R)$ of all continuous functions.
We investigate various sequential convergences related to
the pointwise convergence and the process of completion of
$C(R)$. In particular, we prove that the pointwise
convergence fails to be strict and prove the existence of
the categorical ring completion of $C(R)$ which differs from
$B(R)$.
