In Baby Rudin book, a theorem states that
Suppose f is bounded on [a,b], f has only finitely many points of discontinuity on [a,b], and α is continuous at every point at which f is discontinuous. Then f∈R(α).
Here is the proof given by Rudin:
Let $\varepsilon > 0$ be given. Put $M = \sup \left\vert f(x) \right\vert$, let $E$ be the set of points at which $f$ is discontinuous. Since $E$ is finite and $\alpha$ is continuous at every point of $E$, we can cover $E$ by finitely many disjoint intervals $\left[ u_j, v_j \right] \subset [a, b]$ such that the sum of the corresponding differences $\alpha\left(v_j\right) - \alpha \left( u_j \right)$ is less than $\varepsilon$. Furthermore, we can place these intervals in such a way that every point of $E \cap (a, b)$ lies in the interior of some $\left[ u_j, v_j \right]$.
Remove the segments $\left( u_j, v_j \right)$ from $[a, b]$. The remaining set $K$ is compact. Hence $f$ is uniformly continuous on $K$, and there exists $\delta > 0$ such that $\left\vert f(s) - f(t) \right\vert < \varepsilon$ if $s \in K$, $t \in K$, $\left\vert s-t \right\vert < \delta$.
Now form a partition $P = \left\{ x_0, x_1, \ldots, x_n \right\}$ of $[a, b]$, as follows: Each $u_j$ occurs in $P$. Each $v_j$ occurs in $P$. No point of any segment $\left( u_j, v_j \right)$ occurs in $P$. If $x_{i-1}$ is not one of the $u_j$, then $\Delta \alpha_i < \delta$.
Note that $M_i - m_i \leq 2M$ for every $i$, and that $M_i - m_i \leq \varepsilon$ unless $x_{i-1}$ is one of the $u_j$. Hence, $$ U(P, f, \alpha) - L(P, f, \alpha) \leq \left[ \alpha(b) - \alpha(a) \right] \varepsilon + 2M \varepsilon.$$
Now my question is can we assume instead of finite set of discontinuities can we assume it to be countable set.
My approach is since α is continuous we can choose $u_j$,$v_j$ such that $\alpha\left(v_j\right) - \alpha \left( u_j \right)$<$\varepsilon$/$2^j$. Hence sum of the corresponding differences $\alpha\left(v_j\right) - \alpha \left( u_j \right)$ is less than $\varepsilon$.
Is this correct approach or I am missing something?