Suggested languages for you:

Americas

Europe

Problem 187

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 \([\mathrm{a}, \mathrm{b}]\).

Expert verified

To prove that a continuous function \(f(x)\) in \([a, b]\) is Riemann integrable, follow these steps:
1. Understand that a continuous function on \([a, b]\) is uniformly continuous, so for every \(\varepsilon>0\), there is a \(\delta>0\) for which \(|x-y|<\delta\) implies \(|f(x)-f(y)|<\frac{\varepsilon}{b-a}\).
2. Divide the interval \([a, b]\) into equally spaced subintervals using a partition with \(\Delta x = \frac{b-a}{n}<\delta\).
3. Find the minimum and maximum values of \(f(x)\) in each subinterval.
4. Calculate the lower sums \(L\) and upper sums \(U\) using these minimum and maximum values.
5. Show that \(U-L<\varepsilon\) by observing that \(U-L \leq \Sigma \frac{\varepsilon}{b-a} * \Delta x =\varepsilon\).
6. Conclude that the continuous function \(f(x)\) is Riemann integrable, as we have shown that there exists a partition with upper and lower sums \(\mathrm{U}\) and \(\mathrm{L}\) such that \(U-L<\varepsilon\).

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 7

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.

Chapter 7

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.

Chapter 7

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 \(\mathrm{g}\).

Chapter 7

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 $b_{a} f d X_{c}=0 \quad c \in(a, b)$ $$ \begin{array}{ll} \text { and }=-\mathrm{f}(\mathrm{a}) & \mathrm{c}=\mathrm{a} \\ \text { and }=\mathrm{f}(\mathrm{b}) & \mathrm{c}=\mathrm{b} \text { . } \end{array} $$

Chapter 7

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 $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{c} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}+\mathrm{b} \int_{\mathrm{c}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} $$

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner