1. hw2 Deadline: 09/30/5:00pm
(1) Open set Property (b) in Theorem 9.3 (page 63): “If U, V are open subsets of Rn, then, U ∩ V is also open.” Let U1, · · · , Um be open sets in Rnwhere m is a natural number. Use mathematical induction and the open set property (b) to prove that U1∩ · · · ∩ Um is also open.
(2) Let S = {(x, y) ∈ R2 : xy ≥ 1}. Find int(S). You need to prove that the set int(S) you found is the right one. Conclude that S is not open by showing S 6= int(S).
(3) True or false. If the statement is true, prove it; if not, give a counterexample.
(a) If A and B are subsets of Rn, is int(A) ⊂ int(B) when A ⊂ B?
(b) Is it true that int(A) ∩ int(B) = int(A ∩ B)?
(4) Sketch D1 and D2and find the boundary sets of D1 and D2. You need to prove that
∂D1 and ∂D2 you found is correct.
(a) D1 = {(x, y) ∈ R2 : x2− y2> 1}.
(b) D2 = {(x, y) ∈ R2 : x ≤ y}.
(5) Let D = {(x, y) ∈ R2 : x2+ 4y2 ≤ 4} and define f : D → R by f (x, y) =p
4 − x2− 4y2. (a) Find ∂D and int(D).
(b) The point p(0, 1) belongs to both of D and ∂D. Is it make sense to you to define/compute fx(0, 1) and fy(0, 1)? What happen when we compute
f (0, 1 + h) − f (0, 1) h
for h > 0? (We have discussed several examples in class. Please do one by yourself).
(c) Let U be the subset of D consisting of points (x, y) so that fx(x, y) and fy(x, y) both exist. In other words,
U = {(x, y) ∈ D : fx(x, y) and fy(x, y) both exist.}.
Find U. Is U open? If not, find int(U ).
(6) Define a function f : R2→ R by
f (x, y) =
xy(x2− y2)
x2+ y2 if (x, y) ∈ R2\ (0, 0) 0 if (x, y) = (0, 0).
(a) Show that fx(0, 0) = lim
h→0
f (h, 0) − f (0, 0)
h , fy(0, 0) = lim
h→0
f (0, h) − f (0, 0) h
both exist.
(b) Let U be the set of all points (x, y) in R2 such that both fx(x, y) and fy(x, y) exist; find U and evaluate fx(x, y) and fy(x, y) for all (x, y) ∈ U. Is U open?
(c) Show that fxy(0, 0) = lim
h→0
fx(0, h) − fx(0, 0)
h , fyx(0, 0) = lim
h→0
fy(h, 0) − fy(0.0) h
both exist. Is it true that fxy(0, 0) = fyx(0, 0)?
1
