Why is restricting the x values to a closed neighborhood important?

In summary, when trying to prove that a function f(x) is continuous, it is not valid to use an x value in the epsilon-delta definition. Instead, we can restrict the x values to a closed neighborhood around a point x = x0 and let epsilon rely on the endpoints. This is because epsilon should be a constant, but delta times x + x0 is not a constant. To ensure that the function is continuous, we can choose a delta that is smaller than 1 and the value of x + x0.
  • #1
JG89
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Say f(x) = x^2 - 1 and I'm trying to prove that f is continuous, then I was told I CANNOT do this:

[tex] |x^2 - x_0^2| = |x-x_0||x+x_0| < \delta|x+x_0| = \epsilon [/tex]

because then our epsilon is relying on an x value. I was told I could restrict the x values to a closed neighborhood about the point x = x_0, and let epsilon rely on the end points, but I cannot let it rely on any x values like in the example shown.

Is this correct? And if it is, why is it?
 
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  • #2
Yes, it's true. It's true because [itex]\epsilon[/itex] is supposed to be a constant and [itex]\delta|x+ x_0|[/itex] is not a constant. so "[itex]\delta|x+ x_0|= \epsilon[/itex] makes no sense.

What you can do is this: Since we only need look at x close to x0, assume |x- x0|< 1. The -1< x- x0< 1 so, adding 2x0 to each part, -1+ 2x0< x+ x_0< 1+ 2x0. The larger of those two numbers, in absolute value, is 1+ 2x0 so [itex]\delta|x+ x_0|< \delta(1+ 2x_0[/itex]. We must choose [itex]\delta[/itex] such that [itex]\delta(1+ 2x_0)< \epsilon[/itex]. Actually, because of the requirement that |x- x0|< 1, we must choose [itex]\delta[/itex] to be the smaller of that number and 1.
 

Related to Why is restricting the x values to a closed neighborhood important?

1. What is modulus of continuity?

Modulus of continuity is a mathematical concept used to measure the smoothness of a function. It represents the maximum distance between the function and its approximation at a given point.

2. How is modulus of continuity calculated?

The modulus of continuity is calculated by taking the supremum (or maximum) of the absolute value of the difference between the function and its approximation over a given interval.

3. What is the significance of finding modulus of continuity?

Finding the modulus of continuity can help determine the accuracy of a function's approximation. It can also provide insight into the behavior and smoothness of a function.

4. What are some methods for finding modulus of continuity?

Some common methods for finding modulus of continuity include using the definition of modulus of continuity, using the Weierstrass approximation theorem, and using the Cauchy mean value theorem.

5. Can modulus of continuity be used for all types of functions?

Yes, modulus of continuity can be used for all types of functions, including continuous functions, piecewise continuous functions, and even functions with discontinuities.

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