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