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