Mathematical Analysis Fundamentals

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