Given $a
$$\alpha(x)=\begin{cases} \alpha(a),&\text{if }a\le x Let $f:[a,b]\to\Bbb R$ continuous. Prove that $f\in R(\alpha)$ and that $$\int_a^bf(x) d\alpha(x)=f(c)\big[\alpha(b)-\alpha(a)\big]$$ My attempt Since $f$ is continuous on $[a, b]$ then $ f \in R (\alpha) $ on $[a,b]$ Then for a partition $P$ such that $a=x_{0} We have $U(P,f,\alpha)=M_{i}\big(\alpha(b)-\alpha(a)\big)$ and $L(P,f,\alpha)=m_{i}\big(\alpha(b)-\alpha(a)\big)$ Can anyone help me to conclude the result please? Thanks for your help