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

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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### 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.
Transcribed Image Text:### 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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