Domain of f(x,y): Open or Closed?

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In summary, the conversation discusses the determination of whether the domain of the function is an open set or closed set and if it is bounded. The book states that it is closed and unbounded, which raises the question of how it can be closed if it goes to infinity. The concept of "closed" is clarified, explaining that it does not necessarily mean that the set is bounded or that every point is an interior point. The conversation also touches on the use of parentheses in intervals and the convention that infinity is not considered to belong to any interval.
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Yankel
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Hello all,

I am trying to determine if the domain of the function:

\[f(x,y)=\frac{\sqrt{ln(x^{2}+y^{2}+1)}}{\left | x \right |+\left | y \right |+\sqrt[4]{xy-1}}\]

Is an open set or closed set and if it's bounded.

The domain is in the attached graph.

View attachment 2468

The book say it is closed and unbounded. I wonder, how can it be closed, when it goes to infinity ?

I may be confusing boundary with open/close, but shouldn't it be open if it goes to infinity, or is it enough to say that since every point is interior it is closed ?

thanks !

Edit: What I mean is, isn't it like sets of 1 variables, were we always write [a,infinity) since infinity can't be closed ?
 

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  • #2
You need to review the definition of "closed". It does not mean that the set is bounded or that every point is an interior point. For example, the whole plane is closed.

Yankel said:
What I mean is, isn't it like sets of 1 variables, were we always write [a,infinity) since infinity can't be closed ?
Interesting remark. A parenthesis as opposed to a square bracket means that the boundary does not belong to the interval. Infinity is not a real point, so by convention it is considered to not belong to any interval. And yes, for a finite interval both boundaries belong to it iff it is closed. However, this does not hold for infinite intervals.
 

Related to Domain of f(x,y): Open or Closed?

1. What is the definition of a "domain" in mathematics?

The domain of a mathematical function is the set of all possible values for the independent variable(s) that can be input into the function to produce an output.

2. How do you determine if a function's domain is open or closed?

A function's domain is considered open if it does not include its boundary points and closed if it includes its boundary points. In other words, if the function is defined at every point within a given interval, the domain is closed. If the function is not defined at one or more points within a given interval, the domain is open.

3. Can a function have both an open and closed domain?

No, a function can only have either an open or closed domain. It cannot have both. A function can have a closed domain and an open range, or vice versa, but not both an open and closed domain.

4. How does the openness or closedness of a domain affect a function's continuity?

The openness or closedness of a domain can affect a function's continuity in the sense that if the domain is open, there may be gaps in the function and it may not be continuous at all points. On the other hand, if the domain is closed, the function will be continuous at all points within the domain.

5. Are there any special cases where a function's domain is neither open nor closed?

Yes, there are some special cases where a function's domain is neither open nor closed. For example, if a function has a removable discontinuity at a certain point, the domain of the function can be considered neither open nor closed. In this case, the function is undefined at that point, but it can be extended to have a closed domain by filling in the gap with the limit of the function at that point.

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