Mark all statements that are correct (there might be more than one statement that is correct). 0 b f³fz fº f> f then f is Riemann integrable over [a,b]. Let f:[a, b] →R be bounded. If Let f:[0,1] → R be given by f(x) = x. For every ɛ>0 there is a partition P ={ Σ(M,(f) – m, (f) ) A x,<ɛ. j=1 0 = 1 The function f:[0,1]→R be given by f(x) = X if x=0 if 0
Mark all statements that are correct (there might be more than one statement that is correct). 0 b f³fz fº f> f then f is Riemann integrable over [a,b]. Let f:[a, b] →R be bounded. If Let f:[0,1] → R be given by f(x) = x. For every ɛ>0 there is a partition P ={ Σ(M,(f) – m, (f) ) A x,<ɛ. j=1 0 = 1 The function f:[0,1]→R be given by f(x) = X if x=0 if 0
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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![### Educational Website Transcription: Understanding Riemann Integrability
**Instructions:** Review the statements below and identify any that are correct. Note that more than one statement may be correct.
1. **Statement 1**
Let \( f:[a,b] \rightarrow \mathbb{R} \) be a bounded function. If
\[
\int_a^b \overline{f} \geq \int_a^b \underline{f}
\]
then \( f \) is Riemann integrable over \([a,b]\).
2. **Statement 2**
Let \( f:[0,1] \rightarrow \mathbb{R} \) be given by \( f(x) = x \). For every \( \varepsilon > 0 \), there exists a partition \( P = \{ x_0, x_1, \ldots, x_n \} \) of \([0,1]\), such that
\[
\sum_{j=1}^{n} \left( M_j(f) - m_j(f) \right) \Delta x_j < \varepsilon.
\]
3. **Statement 3**
The function \( f:[0,1] \rightarrow \mathbb{R} \) is given by
\[
f(x) =
\begin{cases}
0 & \text{if } x = 0 \\
\frac{1}{x} & \text{if } 0 < x \leq 1
\end{cases}
\]
is Riemann integrable.
4. **Statement 4**
Let \( P \) be a partition of \([a,b]\) and \( Q = P \cup \{c\} \), where \( c \in [a,b] \), \( c \notin P \). Then \( L(f, Q) \leq L(f, P) \) and \( U(f, P) \geq U(f, Q) \).
5. **Statement 5**
Let \( P, Q \) be partitions of \([a,b]\) and \( f:[a,b] \rightarrow \mathbb{R} \) be bounded. Then \( U(f, P) \leq L(f,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fe09fdc95-efca-4d4a-be31-f607b95829c4%2Fg2r7b67_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Transcription: Understanding Riemann Integrability
**Instructions:** Review the statements below and identify any that are correct. Note that more than one statement may be correct.
1. **Statement 1**
Let \( f:[a,b] \rightarrow \mathbb{R} \) be a bounded function. If
\[
\int_a^b \overline{f} \geq \int_a^b \underline{f}
\]
then \( f \) is Riemann integrable over \([a,b]\).
2. **Statement 2**
Let \( f:[0,1] \rightarrow \mathbb{R} \) be given by \( f(x) = x \). For every \( \varepsilon > 0 \), there exists a partition \( P = \{ x_0, x_1, \ldots, x_n \} \) of \([0,1]\), such that
\[
\sum_{j=1}^{n} \left( M_j(f) - m_j(f) \right) \Delta x_j < \varepsilon.
\]
3. **Statement 3**
The function \( f:[0,1] \rightarrow \mathbb{R} \) is given by
\[
f(x) =
\begin{cases}
0 & \text{if } x = 0 \\
\frac{1}{x} & \text{if } 0 < x \leq 1
\end{cases}
\]
is Riemann integrable.
4. **Statement 4**
Let \( P \) be a partition of \([a,b]\) and \( Q = P \cup \{c\} \), where \( c \in [a,b] \), \( c \notin P \). Then \( L(f, Q) \leq L(f, P) \) and \( U(f, P) \geq U(f, Q) \).
5. **Statement 5**
Let \( P, Q \) be partitions of \([a,b]\) and \( f:[a,b] \rightarrow \mathbb{R} \) be bounded. Then \( U(f, P) \leq L(f,
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