# Functions of several variables

1. Mar 14, 2012

### miglo

1. The problem statement, all variables and given/known data
so im supposed to find the domain of the function, the range, describe its level curves, find the boundary of the functions domain, determine if the boundary is an open region, a closed region, or neither, and decide it the domain is bounded or unbounded.
f(x,y)=sqrt(y-x)
and f(x,y)=xy

2. Relevant equations

3. The attempt at a solution
ive found the domains and range for the function, described the level curves and found the boundary of the domain for the first function but what is really confusing me is whether the domain is open of closed.
for the first one the domain is all points in the xy-plane such that y is greater than or equal to x, at first i thought the domain was closed but some tutors told me that its actually open and closed. can anyone explain this to me as the tutors themselves did not know how to explain it.
for the second function i know the domain is all points in the xy-plane, i went ahead and took a peak at the answers and it says the domain is open and closed, i dont understand this, my first guess was that it was just open but i was wrong
my professor never mentioned that it could be both open and closed, so i dont understand it

2. Mar 15, 2012

### Staff: Mentor

The domain of the first function is the half-plane consisting of the line y = x and all of the points above this line. Since the boundary of this set is the line y = x, and the boundary is part of the set, the domain is closed. See http://en.wikipedia.org/wiki/Closed_set.

Per the article, the set [1, ∞) in R is closed, and from this we can reasonably conclude that {(x, y) | y ≥ x} is also closed. On the other hand, the set [0, 1) is neither open nor closed.

Some sets are both open and closed; e.g., the real line (-∞, ∞) and the plane R2.
I think you took a peek at the answers. Peaks are generally too large to take.

3. Mar 15, 2012

### miglo

thanks!

and yeah i took a peek not a peak haha

4. Mar 15, 2012

### Staff: Mentor

Or you could say, in a fit of pique you took a peek at the peak.