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According to Example 5.9, it has a countably infinite number of discontinuities at rational numbers of $\left[0\text{,}1\right]$ (the exception is only $x=0$). To show that $f\in R\left(a\text{,}b\right)$ note that ${S}_{\ast }\left(f\text{,}\mathcal{P}\right)=0$ for every partition $\mathcal{P}$ of $\left[0\text{,}1\right]$. Therefore, ${S}_{\ast }\left(f\right)=0$ and it remains to ...

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