Epsilon-Delta continuity proof

In summary, the conversation discusses how to obtain values for δ, x, and |x^2+x+3| in different scenarios. The first two questions focus on obtaining a value for δ, with the first question specifically asking about δ<1 and the second question inquiring about x∈(0,2). The final question involves finding values for |x^2+x+3|, with the use of the absolute value notation. The conversation also mentions that delta is defined to be between 0 and min(1, epsilon/9), where min(x, y) represents the minimum of x and y. The conversation ends with a request for clarification on what values x can take when delta < 1 and |x
  • #1
gomes.
58
0
http://img34.imageshack.us/img34/1989/analysis123523456.jpg


I'm trying to work through some examples, but I am not sure where the following comes from:



1. circled in black -- how do i get the δ<1?



2. circled in red -- how do I get 0<x<2, i.e. x∈(0,2)?



3. cirlced in blue -- how do i get |x^2+x+3|<9



Thanks, most appreciated.
 
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  • #2
Let's start with 1 and 2.

delta is defined to be between 0 and min(1, epsilon/9).
min(x, y) means the minimum of x and y.
This means that delta < 1 and perhaps a lot smaller.

With delta < 1 and |x-1| < delta, what values can x take?
 

FAQ: Epsilon-Delta continuity proof

What is an Epsilon-Delta continuity proof?

An Epsilon-Delta continuity proof is a method used in mathematical analysis to rigorously prove that a function is continuous at a given point. It involves using two variables, epsilon and delta, to quantify the distance between inputs and outputs of a function.

How does an Epsilon-Delta continuity proof work?

An Epsilon-Delta continuity proof works by showing that for any positive value of epsilon, there exists a corresponding positive value of delta such that the distance between the input and output of a function is less than epsilon whenever the distance between the input and a given point is less than delta.

What is the importance of an Epsilon-Delta continuity proof?

An Epsilon-Delta continuity proof is important because it provides a rigorous and precise way to prove that a function is continuous at a given point. This is essential in many areas of mathematics and science, as the concept of continuity is fundamental in understanding the behavior of functions.

What are the key steps in an Epsilon-Delta continuity proof?

The key steps in an Epsilon-Delta continuity proof include defining epsilon and delta, stating the definition of continuity, identifying the given function and point, and setting up the inequality to be proven. Then, using algebra and the properties of inequalities, the proof is completed by showing that the given delta satisfies the definition of continuity.

Are there any tips for understanding and solving Epsilon-Delta continuity proofs?

Some tips for understanding and solving Epsilon-Delta continuity proofs include starting with simple examples, understanding the definition of continuity, and breaking the proof down into smaller steps. It is also helpful to practice with different types of functions and points to gain a better understanding of the process.

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