Delta/Epsilon Proof for General Cases

In summary, a Delta/Epsilon proof for general cases is a method used in mathematical analysis to prove the limit of a function using the concepts of delta and epsilon. It is important because it provides a rigorous and precise way to prove the limit and the key steps include stating the limit definition, choosing appropriate values for delta and epsilon, and writing the proof using the delta-epsilon definition. The delta value is chosen based on the conditions of the limit and a Delta/Epsilon proof can be used for any function as long as it meets the necessary conditions.
  • #1
ssayan3
15
0
Hey guys, this isn't a math problem for homework or for a course, but simply for my own interest... This is one that I couldn't crack back in Analysis but that I'd like to get some help on resolving for my own peace of mind:

Prove f(x,y) = xy is continuous at (x,y) in a domain D[tex]\subset[/tex] R. I can do this easily with specific cases of (x,y) but I couldn't ever figure out the general case :-(
 
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  • #2
Try writing xy - ab = (xy - ay) + (ay - ab) and working with that.
 

FAQ: Delta/Epsilon Proof for General Cases

What is a Delta/Epsilon Proof for General Cases?

A Delta/Epsilon proof for general cases is a method used in mathematical analysis to prove the limit of a function. It involves using the concepts of delta and epsilon, which represent the distance between two points on a graph and the precision of the limit, respectively.

Why is a Delta/Epsilon Proof important?

A Delta/Epsilon proof is important because it provides a rigorous and precise way to prove the limit of a function. It allows for a clear and logical explanation of why a limit exists and what its value is.

What are the key steps in a Delta/Epsilon Proof?

The key steps in a Delta/Epsilon proof include: 1) stating the limit definition, 2) choosing an appropriate delta value, 3) finding a corresponding epsilon value, 4) writing the proof using the delta-epsilon definition, and 5) concluding the proof by showing that the limit holds true for all values within the chosen delta.

How do you choose the delta value in a Delta/Epsilon Proof?

The delta value in a Delta/Epsilon proof is chosen based on the conditions of the limit, such as the function being continuous. It should be small enough to ensure that the distance between the input and the limit is within the specified precision (epsilon) but not so small that it becomes impossible to find an appropriate epsilon value.

Can a Delta/Epsilon Proof be used for any function?

Yes, a Delta/Epsilon proof can be used for any function as long as it meets the necessary conditions, such as being continuous. However, the process may vary slightly depending on the type of function (e.g. polynomial, trigonometric, exponential).

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