### Problem1. Use the ε-δ definition to prove that $ f(x) = x^2 $ is uniformly continuous on $[a, b]$.### ExplanationThe problem asks you to use the epsilon-delta ($ \epsilon-\delta $) definition of uniform continuity to demonstrate that the function $ f(x) = x^2 $ is uniformly continuous over the closed interval $[a, b]$. The uniform continuity of a function $ f $ on an interval is defined as follows:A function $ f $ is uniformly continuous on an interval if, for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for all $ x, y $ in the interval, if $ |x - y| < \delta $, then $ |f(x) - f(y)| < \epsilon $.Your task is to apply this definition to prove the uniform continuity of $ f(x) = x^2 $ on the specified interval.

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1. Use the e-ổ definition to prove that f(x) = x² is uniformly continuous on [a, 이.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

### Problem

1. Use the ε-δ definition to prove that $ f(x) = x^2 $ is uniformly continuous on $[a, b]$.

### Explanation

The problem asks you to use the epsilon-delta ($ \epsilon-\delta $) definition of uniform continuity to demonstrate that the function $ f(x) = x^2 $ is uniformly continuous over the closed interval $[a, b]$.

The uniform continuity of a function $ f $ on an interval is defined as follows:

A function $ f $ is uniformly continuous on an interval if, for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for all $ x, y $ in the interval, if $ |x - y| < \delta $, then $ |f(x) - f(y)| < \epsilon $.

Your task is to apply this definition to prove the uniform continuity of $ f(x) = x^2 $ on the specified interval.

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