Given that a bounded function \(\mathrm{f}(\mathrm{x})\) is Riemann integrable in \([a, b]\) if and only if given any \(\varepsilon>0\) there exists a partition with upper and lower sums \(\mathrm{U}\) and \(\mathrm{L}\) such that \(\mathrm{U}-\mathrm{L}<\varepsilon\), prove that a continuous function \(\mathrm{f}(\mathrm{x})\) in \([\mathrm{a}, \mathrm{b}]\) is Riemann integrable in \([\mathrm{a}, \mathrm{b}]\).

Suppose f is defined on \([0,2]\) as follows: $$ \begin{aligned} &\mathrm{f}(\mathrm{x})=1 \text { for } 0 \leq \mathrm{x}<1 \\ &\text { and }=2 \text { for } 1 \leq \mathrm{x} \leq 2 \text { . } \end{aligned} $$ Show that \(\mathrm{f}\) is Riemann integrable.

Find the derivatives of a) $\mathrm{x} \int_{1} \mathrm{t}^{2} \mathrm{dt} \quad\( with respect to \)\mathrm{x}$. b) ${ }^{\text {(t) } 2} \int_{1} \sin \left(\mathrm{x}^{2}\right) \mathrm{d} \mathrm{x}$ with respect to t.

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 \(\mathrm{g}\).

Let \(\mathrm{f}\) be a function from \([\mathrm{a}, \mathrm{b}]\) into \(\mathrm{R}\) which is continuous at \(c \in[a, b]\) and let \(X_{c}\) be the characteristic function of \(c\), i.e., $$ \begin{array}{cc} X_{c}(x)=1 & x=c \\ \text { and }=0 & x \neq c . \end{array} $$ Show, using the definition of the Stieltjes integral that $b_{a} f d X_{c}=0 \quad c \in(a, b)$ $$ \begin{array}{ll} \text { and }=-\mathrm{f}(\mathrm{a}) & \mathrm{c}=\mathrm{a} \\ \text { and }=\mathrm{f}(\mathrm{b}) & \mathrm{c}=\mathrm{b} \text { . } \end{array} $$

Prove the following: a) If \(\mathrm{f}(\mathrm{x}) \leq \mathrm{g}(\mathrm{x})\) and \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \leq \mathrm{b}_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x} $$ b) If \(\mathrm{f}(\mathrm{x})\) is bounded and Riemann integrable and if \(\mathrm{c}\) is any point such that \(\mathrm{C} \in[\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{c} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}+\mathrm{b} \int_{\mathrm{c}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} $$