Showing a set is closed with the definition of continuity

In summary, the conversation discusses how to prove that the subset A = {(x, y) : xy = 1} in R^2 is closed using the "closed set formulation" of continuity. The attempt at a solution involves using the function f(x, y) = xy, but there is a difficulty with the proof as it is not continuous at x = 0. The conversation concludes with the suggestion to use a different function that is continuous on all of R^2, making the proof almost trivial.
  • #1
radou
Homework Helper
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Homework Statement



I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity.

The Attempt at a Solution



So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is closed.

The set A can be written as A = {(x, 1/x) : x is in R\{0}}. Since f(x) = 1/x is a continuous function on R\{0}, and since any subset of R containing only one element is closed, f^-1{a} = {1/a} is a closed subset of R, for every a in R\{0}. Any ordered pair of the form (a, 1/a) can be written as a cartesian product of the sets {a} x {1/a}, which is closed, since the sets are closed. But an infinite union of such sets need not be closed. I feel it's warm around here, but I just can't figure it out.

Perhaps I'm not on the right track. Any help appreciated, as always.
 
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  • #2
...no thoughts?
 
  • #3
One major difficulty with your proof is that your function is NOT continuous on any set including x= 0! And there is nothing in your initial statement that says you can exclude x= 0.

I would be inclined, instead, to use the function f(x, y)= xy. That is continuous for all (x, y) in R2.
 
  • #4
HallsofIvy said:
I would be inclined, instead, to use the function f(x, y)= xy. That is continuous for all (x, y) in R2.

Thanks a lot. In this case, the proof is almost trivial - since {1} is a closed set in R, its preimage must be closed, and it is exactly the set we're looking at.
 

Related to Showing a set is closed with the definition of continuity

1. What is the definition of continuity?

The definition of continuity is a mathematical concept that describes the behavior of a function at a specific point or over an interval. It states that a function is continuous if the limit of the function at a given point is equal to the value of the function at that point.

2. How do you show that a set is closed using the definition of continuity?

To show that a set is closed using the definition of continuity, you need to prove that for every limit point in the set, the function's limit at that point is equal to the value of the function at that point. This effectively shows that the set contains all of its limit points and is therefore closed.

3. What is a limit point?

A limit point is a point in a set where every neighborhood of that point contains at least one other point in the set. In other words, it is a point where a function approaches as the input approaches a specific value.

4. Can you provide an example of showing a set is closed using the definition of continuity?

Yes, for example, let's say we have a function f(x) = x^2 and we want to show that the set of all real numbers is closed. We can do this by taking any limit point in the set, say x = 2. We can then show that the limit of f(x) as x approaches 2 is equal to the value of f(x) at 2, which is 4. Therefore, the set of all real numbers contains all of its limit points and is closed.

5. Are there any other methods for showing a set is closed besides using the definition of continuity?

Yes, there are other methods such as using the closure property, the sequential characterization of closed sets, or the topological definition of closed sets. However, using the definition of continuity is a common and straightforward method for showing a set is closed.

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