Question: If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$ with integrator $\alpha$.

Quasi-continuous means $f(x_+)$ and $f(x_-)$ exist for all $x \in [a,b]$. Left {Right} continuous means $f(x)=f(x_-)$ {=$f(x_+)$}

Text: Real Analysis by Carothers, also Rudin

I have already shown that I can find a partition $P=\{a=t_0<t_1<...<t_n=b\}$ such that for $1 \le k \le n$ the oscillations $\omega(f,(t_{k-1}, t_k])<\epsilon$.

I've tried using the Riemann criterion using upper and lower sums but I couldn't make any headway.

I can rule out potential ways that it would not be integrable since $f$ and $\alpha$ do not share any common sided discontinuities. Any suggestions or help would be much appreciated. Thanks.


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