Ross Elementary Analysis Epsilon Delta Proofs

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Start date Dec 31, 2014
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Analysis Delta Elementary Epsilon Epsilon delta Proofs

In summary, epsilon delta proofs are used in elementary analysis to formally prove the continuity of a function at a point. This is achieved by finding a relationship between the variables epsilon and delta that satisfies the definition of continuity. There are two types of epsilon delta proofs: one-sided and two-sided, which are used depending on the function and its domain. Epsilon delta proofs cannot be used to prove differentiability and have limitations such as being time-consuming and only applicable to real-valued functions.

Dec 31, 2014

unintuit

Does Ross's book teach and/or use Epsilon-delta proof techniques?

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Dec 31, 2014

bolbteppa

Yes, for example here
http://books.google.ie/books?id=5JxHZNpMq3AC&lpg=PP1&pg=PA37#v=onepage&q&f=false
he gives a huge discussion before actually doing the formal proof, does this with a few more examples, and spent the section before easing into the proofs as well.

1. What is the purpose of epsilon delta proofs in elementary analysis?

The purpose of epsilon delta proofs is to formally prove the continuity of a function at a point. This is done by showing that for any small value of epsilon (ε), there exists a corresponding small value of delta (δ) such that the function values within a delta neighborhood of the point will be within an epsilon neighborhood of the function value at the point.

2. How do you construct an epsilon delta proof?

To construct an epsilon delta proof, you must first define the function, the point at which you are proving continuity, and the variables epsilon and delta. Then, you must use algebraic manipulations to find a relationship between epsilon and delta that satisfies the definition of continuity. Finally, you must show that the chosen value of delta works for all values of epsilon, thus proving continuity at the chosen point.

3. What is the difference between a one-sided and two-sided epsilon delta proof?

In a one-sided epsilon delta proof, the values of delta and epsilon are only considered on one side of the chosen point. This is typically used when the function is only defined on one side of the point, such as in a piecewise function. In a two-sided epsilon delta proof, the values of delta and epsilon are considered on both sides of the point, and the proof must work for both sides in order to show continuity.

4. Can you use epsilon delta proofs to prove differentiability?

No, epsilon delta proofs are only used to prove the continuity of a function at a point. To prove differentiability, you must use the definition of the derivative and show that the limit of the difference quotient exists at the chosen point.

5. Are there any limitations to using epsilon delta proofs?

One limitation of using epsilon delta proofs is that they can be time-consuming and require a lot of algebraic manipulation. Additionally, they can only prove continuity at a specific point, so multiple proofs may be needed to show continuity of a function on an interval. Also, epsilon delta proofs can only be used for real-valued functions, so they cannot be applied to complex-valued functions.