Is f Continuous Everywhere? Analyzing the Limit of a Fractional Function

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In summary, continuity in mathematics refers to a function's smoothness as the input values change, without any sudden jumps or breaks in the graph. To check for continuity at a point, the function must be defined, the limit at that point must exist, and the limit must be equal to the value of the function. Continuity is different from differentiability, as a function can be one but not the other. A function can also be continuous but not uniformly continuous, as uniform continuity requires small changes in output for all input values. The Intermediate Value Theorem, which states that a continuous function must take on every value between two points, helps to determine continuity at a specific point.
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Calcotron
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Well, my first question was answered so I figured I would post the second problem I had problems with. It is:

[tex] f(x) = lim _{n->\infty}\frac{x^{2n} - 1}{x^{2n} + 1}[/tex]

Where is f continuous? My first thought is that it is continuous everywhere since I can't find an x value that would make the bottom part of the fraction 0. Isn't that function 1 at for [tex]\infty < x \leq-1[/tex] or [tex] 1 \leq x < \infty[/tex] and -1 otherwise?
 
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Lol, let's just pretend this question never happened ok?
 
  • #3
I take it that means you've solved the question then?
 

1. What is the definition of continuity in mathematics?

Continuity is a property of a mathematical function where the output values change smoothly as the input values change. It means that there are no sudden jumps or breaks in the graph of the function.

2. How do you check for continuity at a point?

To check for continuity at a point, we need to make sure that the function is defined at that point, the limit of the function at that point exists, and the limit is equal to the value of the function at that point.

3. Is continuity the same as differentiability?

No, continuity and differentiability are two different concepts in mathematics. A function can be continuous at a point but not differentiable, and it can be differentiable at a point but not continuous.

4. Can a function be continuous but not uniformly continuous?

Yes, a function can be continuous but not uniformly continuous. Uniform continuity requires that the change in output values is small for all input values, whereas continuity only requires that the change in output values is small for nearby input values.

5. How does the Intermediate Value Theorem relate to continuity?

The Intermediate Value Theorem states that if a continuous function has two different output values at two points, then it must take on every value in between those two points. This theorem is only applicable for continuous functions, and it helps to show that a function is continuous at a certain point.

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