Let f: RR be given by f(x) = x²-4 X-2 2 if x # 2 . Find the oscillation (2) = inf{o (D(2,8)): 8>0}, where (D(2,8)) = sup{|f(x)-f(y)]: x,y D(2,8)), off at x = 2. if x=2
Let f: RR be given by f(x) = x²-4 X-2 2 if x # 2 . Find the oscillation (2) = inf{o (D(2,8)): 8>0}, where (D(2,8)) = sup{|f(x)-f(y)]: x,y D(2,8)), off at x = 2. if x=2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Question 2**
Let \( f: \mathbb{R} \to \mathbb{R} \) be given by
\[
f(x) =
\begin{cases}
\frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\
2 & \text{if } x = 2
\end{cases}
\]
Find the oscillation \( \omega_f(2) = \inf(\omega_f(D(2,\delta)) : \delta > 0) \), where \( \omega_f(D(2,\delta)) = \sup \{|f(x) - f(y)| : x, y \in D(2,\delta) \} \), of \( f \) at \( x = 2 \).
**Question 3**
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by
\[
f(x) =
\begin{cases}
\left( x - \frac{\pi}{2} \right) \cos\left(\frac{1}{x - \frac{\pi}{2}}\right) & \text{if } x \neq \frac{\pi}{2} \\
0 & \text{if } x = \frac{\pi}{2}
\end{cases}
\]
Find the oscillation \( \omega_f\left(\frac{\pi}{2}\right) = \inf(\omega_f(D(\frac{\pi}{2}, \delta)) : \delta > 0) \), where \( \omega_f(D(\frac{\pi}{2},\delta)) = \sup \{|f(x) - f(y)| : x, y \in D(\frac{\pi}{2},\delta) \} \), of \( f \) at \( x = \frac{\pi}{2} \).
**Question 4**
Let \( f:\mathbb{R} \to \mathbb{R} \) be given by
\[
f(x) =
\begin{cases}
\frac{1}{1+x^2} & \text{if } x \in \mathbb{Q} \\
4 - x^2 & \text{if } x \in \mathbb{R} \backslash \mathbb{Q}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fef4a7e08-81f6-497f-98bc-064c0be722b7%2F8j99j1m_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 2**
Let \( f: \mathbb{R} \to \mathbb{R} \) be given by
\[
f(x) =
\begin{cases}
\frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\
2 & \text{if } x = 2
\end{cases}
\]
Find the oscillation \( \omega_f(2) = \inf(\omega_f(D(2,\delta)) : \delta > 0) \), where \( \omega_f(D(2,\delta)) = \sup \{|f(x) - f(y)| : x, y \in D(2,\delta) \} \), of \( f \) at \( x = 2 \).
**Question 3**
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by
\[
f(x) =
\begin{cases}
\left( x - \frac{\pi}{2} \right) \cos\left(\frac{1}{x - \frac{\pi}{2}}\right) & \text{if } x \neq \frac{\pi}{2} \\
0 & \text{if } x = \frac{\pi}{2}
\end{cases}
\]
Find the oscillation \( \omega_f\left(\frac{\pi}{2}\right) = \inf(\omega_f(D(\frac{\pi}{2}, \delta)) : \delta > 0) \), where \( \omega_f(D(\frac{\pi}{2},\delta)) = \sup \{|f(x) - f(y)| : x, y \in D(\frac{\pi}{2},\delta) \} \), of \( f \) at \( x = \frac{\pi}{2} \).
**Question 4**
Let \( f:\mathbb{R} \to \mathbb{R} \) be given by
\[
f(x) =
\begin{cases}
\frac{1}{1+x^2} & \text{if } x \in \mathbb{Q} \\
4 - x^2 & \text{if } x \in \mathbb{R} \backslash \mathbb{Q}
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