1. Quizz 1 We have studied lim
(x,y)→(0,0)
x2− xy
√x −√
y in class. The set
D = {(x, y) ∈ R2 : x > 0, y > 0, x 6= y}
is the (maximal) domain of the function f (x, y) = x2− xy
√x −√
y. Before understanding the limit, we need to understand the function and its domain. We have learned the notion of open set and would like to determine whether D is open or not. The following exercises will help us to prove that D is open without proving int(D) = D directly.
(1) ( ) The set U1 = {(x, y) ∈ R2 : x > 0} is open.
(2) ( ) The set U2 = {(x, y) ∈ R2 : y > 0} is open.
(3) ( ) The set V1 = {(x, y) ∈ R2 : x > y} is open.
(4) ( ) The set V2 = {(x, y) ∈ R2 : x < y} is open.
(5) ( ) The equality {(x, y) ∈ R2: x 6= y} = V1∪ V2 holds.
(6) ( ) The set U3 = {(x, y) ∈ R2 : x 6= y} is open.
(7) ( ) The equality D = U1∩ U2∩ U3 holds.
(8) ( ) The set D is open.
1
2
The domain of the function f (x, y) =p
1 − x2− y2 is given by D = {(x, y) ∈ R2 : x2+ y2 ≤ 1}.
To analyze the function, it is important for us to study the “topological properties” of D. For example, can we find the extremum (maximum or minimum) of f ? To find the extremum of f, we need to compute the partial derivatives of f. (The method of finding extremum for functions of several variables will be introduced later). The partial derivatives in all direction can be defined only at interior points of D.
The set of all boundary points of D is denoted by ∂D. Recall that a point x ∈ Rn is a boundary point of a set A if B(x, ) ∩ A and B(x, ) ∩ Ac are nonempty for any > 0.
(1) ( ) The boundary set ∂D of D equals {(x, y) ∈ R2 : x2+ y2 = 1}.
(2) ( ) The equality int(D) = {(x, y) ∈ R2: x2+ y2 < 1} holds.
(3) ( ) The set D is not open.
(4) ( ) In general, if a set A of Rn contains a boundary point, then it is never an open set.
(5) ( ) If a subset A of Rncontains no boundary points of A, then A must be open.
