Let $\{\alpha_i\}$ be a sequence of monotonically increasing functions on $[a,b]$.
Suppose $f\in \mathscr{R}(\alpha_i), \forall i\in \mathbb{N}$ and $\sum \alpha_i$ is convergent.
Let $A=\sum_{i=1}^{\infty} \alpha_i$
Then $f\in \mathscr{R}(A)$?
If it is not true, then what if $f$ is cotinuous at every discontinuity of each $\alpha_i$?