3. To show that $ f $ is not uniformly continuous on $ A $, you should demonstrate that there is some $ \varepsilon > 0 $ such that no matter what $ \delta > 0 $ you pick, you can find points $ x, y \in A $ such that $|x - y| < \delta$ and $|f(x) - f(y)| \geq \varepsilon$. Consider the proof below demonstrating that $ f(x) = \frac{1}{x} $ is not uniformly continuous on $ (0, 1) $.**Proof:** Consider $ \varepsilon = 1/2 $. Let $ \delta > 0 $ be given. Then, there exists $ n \in \mathbb{N} $ such that $ \frac{1}{n} < \delta $. Moreover, we can choose $ n \geq 3 $. If $ x = \frac{2}{n} $ and $ y = \frac{1}{n} $ which are in $ (0, 1) $, then\[0 < |x - y| = \left| \frac{2}{n} - \frac{1}{n} \right| = \frac{1}{n} < \delta\]and\[|f(x) - f(y)| = \left|\frac{1}{2/n} - \frac{1}{1/n}\right| = \left|n/2 - n\right| = n/2 \geq 1/2 = \varepsilon.\]Using a similar proof, show that $ f(x) = \frac{1}{x} $ is not uniformly continuous on $ (0, 1) $.

Given: Given function is fx=1x2 We want to show that fx=1x2 is not uniformly continuous on 0,1

3. To show that f is not uniformly continuous on A, you should demonstrate that there is some e> 0 such that no matter what &> 0 you pick, you can find points a, y E A such that r y < 8 and f(r)- f (u) 2 e. Consider the proof below demonstrating that f(a) is not uniformly continuous on (0, 1). Proof: Consider e = 1/2. Let &> 0 be given. Then, there exists n E N such that <6. Moreover, we can choosen 2 3. If r and y which are in (0, 1), then 0 < |a - yl = and IS(*)- ()-/(2/n)- f(1/n)| n/22 1/2 e. Using a similar proof, show that f(a) = is not uniformly continuous on (0, 1).

3. To show that f is not uniformly continuous on A, you should demonstrate that there is some e> 0 such that no matter what &> 0 you pick, you can find points a, y E A such that r y < 8 and f(r)- f (u) 2 e. Consider the proof below demonstrating that f(a) is not uniformly continuous on (0, 1). Proof: Consider e = 1/2. Let &> 0 be given. Then, there exists n E N such that <6. Moreover, we can choosen 2 3. If r and y which are in (0, 1), then 0 < |a - yl = and IS(*)- ()-/(2/n)- f(1/n)| n/22 1/2 e. Using a similar proof, show that f(a) = is not uniformly continuous on (0, 1).

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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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

3. To show that $ f $ is not uniformly continuous on $ A $, you should demonstrate that there is some $ \varepsilon > 0 $ such that no matter what $ \delta > 0 $ you pick, you can find points $ x, y \in A $ such that $|x - y| < \delta$ and $|f(x) - f(y)| \geq \varepsilon$. Consider the proof below demonstrating that $ f(x) = \frac{1}{x} $ is not uniformly continuous on $ (0, 1) $.

**Proof:** Consider $ \varepsilon = 1/2 $. Let $ \delta > 0 $ be given. Then, there exists $ n \in \mathbb{N} $ such that $ \frac{1}{n} < \delta $. Moreover, we can choose $ n \geq 3 $. If $ x = \frac{2}{n} $ and $ y = \frac{1}{n} $ which are in $ (0, 1) $, then

\[
0 < |x - y| = \left| \frac{2}{n} - \frac{1}{n} \right| = \frac{1}{n} < \delta
\]

and

\[
|f(x) - f(y)| = \left|\frac{1}{2/n} - \frac{1}{1/n}\right| = \left|n/2 - n\right| = n/2 \geq 1/2 = \varepsilon.
\]

Using a similar proof, show that $ f(x) = \frac{1}{x} $ is not uniformly continuous on $ (0, 1) $.

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