# Can Riemann integrability imply Riemann-Stieltjes integrability, or vice versa?

That is, is it possible that $f\in R$ but $f\notin R(\alpha)$ for some $\alpha$ other than $\alpha(x)=x$, or $f\in R(\alpha)$ for some $\alpha$ other than $\alpha(x)=x$, but $f\notin R$? If it is, can you give me an example? Thanks.