A question about uniform continuity (analysis)

In summary, a different proof was proposed for question 19.2 in the given link, using the equation |f(x)-f(y)| = |x-y||x+y| < \delta|x+y|, with the largest value of |x+y| being 6. By setting \delta = \epsilon/6, the proof was shown to be valid for all possibilities. There was a minor error with a < symbol, but overall the proof was deemed correct.
  • #1
Artusartos
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Homework Statement



For question 19.2 in this link:

http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw7sum06.pdf [Broken]

I came up with a different proof, but I'm not sure if it is correct...

Homework Equations





The Attempt at a Solution



Let [tex]|x-y|< \delta[/tex]

For [tex]|f(x)-f(y)| = |x^2 - x^y| = |x-y||x+y| < \delta|x+y|[/tex], we know that the largest that |x+y| can be is 6. So if we let [tex]\delta= \epsilon/6 [/tex]...

We will have

[tex] |f(x)-f(y)| = |x^2 - x^y| = |x-y||x+y| < \delta|x+y| < (\epsilon/6)(6) = \epsilon [/tex]

If this is true for the largest possibility, then it must be possible for all of them...

Do you think my answer is correct, or is there something that I'm missing?

Thanks in advance
 
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  • #2
The last < should be a <=, but apart from that it is fine.
 

What is uniform continuity?

Uniform continuity is a concept in mathematical analysis that describes how a function behaves at different points in its domain. It is a stronger version of continuity, where the function's behavior at different points is consistent and does not depend on the size of the interval between the points.

What is the difference between uniform continuity and continuity?

The main difference between uniform continuity and continuity is that continuity only requires the function to be continuous at each point in its domain, while uniform continuity requires the function to be continuous at each point and also have a consistent behavior across the entire domain.

How is uniform continuity related to differentiability?

Uniform continuity and differentiability are closely related, as both are properties that describe how a function behaves at different points in its domain. However, differentiability is a stronger condition that requires the function to have a well-defined derivative at each point in its domain, while uniform continuity does not have this requirement.

What are the main applications of uniform continuity?

Uniform continuity has many applications in mathematics, particularly in the areas of calculus, analysis, and differential equations. It is also used in physics, engineering, and other fields to model and analyze real-world phenomena.

How can uniform continuity be proven or disproven for a given function?

To prove uniform continuity for a given function, one can use the definition of uniform continuity and show that the function satisfies it. On the other hand, to disprove uniform continuity, one can find a counterexample or use a theorem that states a condition under which a function is not uniformly continuous.

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