How Can I Prove Continuity Using Epsilon Delta Definition?

In summary, the conversation is about the use of the epsilon delta definition for continuity in advanced calculus. The speaker used the assumption that delta<1 to show that x^4 satisfies the epsilon delta condition, but is unsure why this assumption is allowed. They are seeking clarification or resources on this topic. Another person explains that it is acceptable to assume delta<1 because the definition only applies in a small neighborhood and using a smaller value for delta does not change the overall result.
  • #1
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Recently in adv calc we have been dealing with the epsilon delta definition for continuity, and my professor said that it is ok to assume that delta<1. I actually used this to show that x^4 satisfies the epsilon delta condition but I'm not quite sure why we can take delta<1. I am sure you guys know the definition and it doesn't restrict delta at all.

I'm hoping someone can refer me to a resource or just give me some kind of insight.

Thanks
 
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  • #2
Well, it rests on the simple fact that if |x-y|<min(delta,1), then |x-y|<delta.
 
  • #3
Because you only care about it in some arbitrarily small neighborhood.

You could assume delta < .1 and base a proof off it.
 

1. What is Epsilon Delta?

Epsilon Delta is a mathematical concept used in the study of limits and continuity in calculus.

2. What is the purpose of Epsilon Delta?

The purpose of Epsilon Delta is to provide a rigorous way to define and prove the existence of a limit.

3. How is Epsilon Delta used in calculus?

Epsilon Delta is used to formally define the concept of a limit and to prove the continuity of a function at a given point.

4. What is the difference between Epsilon Delta and other methods of finding limits?

Epsilon Delta is a more formal and rigorous method compared to other methods, such as the limit laws or L'Hopital's rule. It relies on logical and mathematical reasoning rather than intuitive or algebraic manipulations.

5. How can I become proficient in using Epsilon Delta?

To become proficient in using Epsilon Delta, it is important to understand the underlying concepts of limits and continuity in calculus. Practicing with various examples and exercises can also help improve proficiency in using Epsilon Delta.

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