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

[Yankel]

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 ?

 

[Evgeny.Makarov]

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.

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.