What does this answer mean? (continuous functions)

In summary, the conversation discusses working on problems involving vectors, specifically the function f(x) = [-x1/|X|3, -x2/|X|3]. The goal is to find the largest interval of existence, which is denoted as E = R2 ~ {0}. It is understood that this means every point in ##\mathbb{R}^2## except the origin, which is not defined due to the denominator. The notation could also be written as ##\mathbb{R}^2 \setminus \{(0,0)\}##. This set is not considered an interval.
  • #1
spaderdabomb
49
0
Working on some problems that have vectors, for example

f(x) = [-x1/|X|3, -x2/|X|3]

And then I am asked to find the largest interval of existence. The answer says "E = R2 ~ {0}.

I'm not sure what this means. Does it mean the interval of existence is everywhere except 0? Is that what the ~ {0} means? It seems like the answer is that any x1, x2 value in R2 space satisfies continuity, but then I have no idea where the ~{0} comes from.
 
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  • #2
Yes, given the context I'm sure it means "every point in ##\mathbb{R}^2## except the origin. A more standard notation would be ##\mathbb{R}^2 \setminus \{(0,0)\}##. By the way, this set isn't an interval at all.

The reason ##(0,0)## is excluded is because your function is not even defined there because of the ##|x|^3## in the denominator.
 

What does it mean for a function to be continuous?

A continuous function is one that has no sudden changes or breaks in its graph. This means that the function can be drawn without lifting the pencil from the paper.

How can I determine if a function is continuous?

A function is continuous if it satisfies three conditions: 1) The function is defined at every point in its domain, 2) The limit of the function exists at every point in its domain, and 3) The limit of the function at any given point is equal to the function value at that point.

What is the difference between a continuous and a discontinuous function?

A continuous function is one that has a smooth, unbroken graph, while a discontinuous function has a graph that has sudden changes or breaks. This means that a discontinuous function cannot be drawn without lifting the pencil from the paper.

Why is continuity important in mathematics?

Continuity is important in mathematics because it allows us to model real-world situations and make accurate predictions. It also allows us to use powerful tools such as the Intermediate Value Theorem and the Mean Value Theorem to solve problems and prove theorems.

Can a function be continuous at some points and discontinuous at others?

Yes, a function can be continuous at some points and discontinuous at others. This is known as a piecewise continuous function, where different pieces of the function are continuous on different intervals. However, for a function to be considered continuous overall, it must be continuous at every point in its domain.

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