Let \(\mathrm{f}(\mathrm{x})=1, \mathrm{x}\) rational and \(=0, \mathrm{x}\) rational be defined in the interval \([a, b]\). Show that is not Riemann integrable

Proceed from the defintion of the Stieltjes integral to show that the function f given by $$ \begin{array}{cc} \mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})=0 & 0 \leq \mathrm{x} \leq 1 \\ \text { and }=1 & 1<\mathrm{x} \leq 2 \end{array} $$ is not Stieltjes integrable with respect to \(\mathrm{g}\).

Evaluate the Stieltjes integral $^{1} \int_{-1} \mathrm{x} \mathrm{d}|\mathrm{x}|$.

Find \(\mathrm{F}^{\prime}(\mathrm{x})\) where a) \(F(x)=x \int_{0} e^{-(x) 2(t) 2} d t\) b) \(F(x)=\sin x \int_{(x) 2}\left(x^{2}-t^{2}\right)^{n} d t\)

Given that \(\mathrm{f}_{1}(\mathrm{x})\) and \(\mathrm{f}_{2}(\mathrm{x})\) are Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) and that $\mathrm{I} \geq \mathrm{J}$ (where I and J represent the upper and lower integrals, respectively, of any Riemann integrable function), prove that \(\mathrm{f}_{1}(\mathrm{x})+\mathrm{f}_{2}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) and that $\mathrm{b} \int_{\mathrm{a}}\left[\mathrm{f}_{1}(\mathrm{x})+\mathrm{f}_{2}(\mathrm{x})\right] \mathrm{d} \mathrm{x}=\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{1}(\mathrm{x}) \mathrm{dx}+\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{2}(\mathrm{x}) \mathrm{dx}$

a) Prove that if \(\mathrm{f}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then \(\mid \mathrm{f}(\mathrm{x})\) is Riemann integrable on the same interval. b) Prove that if \(\mathrm{f}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\left|\leq \mathrm{b} \int_{\mathrm{a}}\right| \mathrm{f}(\mathrm{x}) \mid \mathrm{d} \mathrm{x} $$