State the definition of a Riemann sum and the definition of what it means for a function to be Riemann integrable.
State the definition of a Riemann sum and the definition of what it means for a function to be Riemann integrable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. State the definition of a Riemann sum and the definition of what it means for a function to be Riemann integrable.
2. Let \( S(f, P) \) denote the upper Darboux sum of \( f \) with partition \( P \). Prove that if \( Q \) is a refinement of \( P \), then \( S(f, Q) \leq S(f, P) \).
3. Let \( f : [0, 1] \rightarrow \mathbb{R} \) be defined by
\[
f(x) =
\begin{cases}
0 & \text{if } x \leq \frac{1}{2} \\
1 & \text{if } x > \frac{1}{2}.
\end{cases}
\]
Let \( \epsilon > 0 \). Prove that there exists a partition, \( P \), of \([0, 1]\) with
\[
S(f, P) - s(f, P) < \epsilon.
\]
What can you conclude about \( f \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F7b42d1f0-fa15-4a3a-b677-3b7abd6ee4dc%2Fpp2c6g9_processed.png&w=3840&q=75)
Transcribed Image Text:1. State the definition of a Riemann sum and the definition of what it means for a function to be Riemann integrable.
2. Let \( S(f, P) \) denote the upper Darboux sum of \( f \) with partition \( P \). Prove that if \( Q \) is a refinement of \( P \), then \( S(f, Q) \leq S(f, P) \).
3. Let \( f : [0, 1] \rightarrow \mathbb{R} \) be defined by
\[
f(x) =
\begin{cases}
0 & \text{if } x \leq \frac{1}{2} \\
1 & \text{if } x > \frac{1}{2}.
\end{cases}
\]
Let \( \epsilon > 0 \). Prove that there exists a partition, \( P \), of \([0, 1]\) with
\[
S(f, P) - s(f, P) < \epsilon.
\]
What can you conclude about \( f \)?
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