Continuity, but is it limits or delta-epsilon or neighborhood?

In summary, the function defined as f(x) = 1/(x^2-x) is discontinuous at all points except for 0 and 1, where it is defined piecewise as f(0) = f(1) = 1. This is because the function has asymptotes at x=0 and x=1, making it impossible to use limit-based or delta-epsilon continuity methods. The function is scatter-shot across R2 and is not continuous anywhere. However, for a smaller domain space, the definition of continuity still applies and the function can be continuous at certain points, such as the rational numbers.
  • #1
filter54321
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EDIT: My presentation of this was pretty bad so I'm trying again.

FIND ALL POINTS OF DISCONTINUITY (IF ANY)

f: {0}U{1/N} --> R
Where N is a natural number

Defined piecewise:
f(x) = 1/(x^2-x)
f(0)=f(1)=1

I'm scared of this problem. Obviously, the function blows up with asymptotes at x=0,1 so plugging the holes piecewise with f(0)=f(1)=1 doesn't help with continuity.

Last semester we covered limit-based continuity and delta-epsilon continuity. However, every single problem went from the real line to the real line. I don't see how you can do continuity with any of the three methods because of the discrete domain. The function doesn't have domain points to generate any of the range points that I want to be less than epsilon close.

The graph of the function is going to be scatter-shot across R2 where every point is floating on its own...I can only concluded that the function isn't continuous ANYWHERE. How do you tackle this one?

Thanks in advance.
 
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  • #2
The definition of continuity hasn't changed for this smaller domain space; you just need to pay closer attention to the exact words.

Let [tex]A\subset\mathbb{R}[/tex]. Then [tex]f: A \to \mathbb{R}[/tex] is continuous at [tex]a \in A[/tex] if, for every [tex]\varepsilon > 0[/tex], there exists [tex]\delta > 0[/tex] such that, whenever [tex]x \in A[/tex] satisfies [tex]|x - a| < \delta[/tex], then [tex]|f(x) - f(a)| < \varepsilon[/tex].

To give an example: the function [tex]\iota: \mathbb{Q} \to \mathbb{R}[/tex] defined by [tex]\iota(x) = x[/tex] is continuous at every point of its domain [tex]\mathbb{Q}[/tex].

You cannot speak of a function being continuous or discontinuous at a point where it's not defined, so in your case, you only have to answer whether the points [tex]0[/tex] and [tex]1/n[/tex] are points of continuity.
 

1. What is continuity in mathematics?

Continuity is a fundamental concept in mathematics that describes the smoothness and connectedness of a function. It means that a function has no breaks, jumps, or holes in its graph and that small changes in the input result in small changes in the output.

2. What is the difference between limits and continuity?

Limits and continuity are closely related concepts, but they are not the same. Continuity refers to the behavior of a function at a specific point, while limits describe the behavior of a function as the input approaches a certain value. In other words, continuity is about the behavior of a function at a point, while limits are about the behavior near a point.

3. What is the delta-epsilon definition of continuity?

The delta-epsilon definition of continuity is a mathematical way of defining continuity. It states that a function f(x) is continuous at a point x=a if for any given value of epsilon (ε), there exists a corresponding delta (δ) such that if the distance between x and a is less than delta (|x-a| < δ), then the distance between f(x) and f(a) is less than epsilon (|f(x)-f(a)| < ε).

4. How is continuity related to the neighborhood of a point?

The neighborhood of a point is a set of all points that are close to that point. In mathematics, a function is continuous at a point if and only if it is defined at that point and the limit of the function exists and is equal to the value of the function at that point. This means that the function behaves similarly in a small neighborhood around that point.

5. Why is continuity important in mathematics and science?

Continuity is important in mathematics and science because it allows us to study and understand the behavior of functions in a smooth and connected manner. It is also essential in many real-world applications, such as physics, engineering, and economics, where continuous functions are used to model and predict natural phenomena. Additionally, the concept of continuity is fundamental in calculus, which is a crucial tool in many scientific fields.

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