Math 2111 Advanced Calculus (I)
Homework 4 Hand in Problems:
2, 3, 4, 6, 7 Lecture Note: 2, 3, 5
1. Let f : [a, b] → R be a continuous function. If f is not a constant function, prove that the range of f is a closed interval.
2. Suppose that S1, S2,· · · , Sn are sets in R and S =S1∪S2∪ · · · ∪Sn. Define Bi = supSi for i= 1,2,· · · , n.
(a) Show that supS= max(B1, B2,· · · , Bn).
(b) IfS is the union of an infinite collection of{Si}, find the relation between supS and {Bi}.
3. LetS1, S2 be sets inR. DefineS ={x |x=x1+x2, wherex1 ∈S2, x2 ∈S2}. Find supS in terms of supS1 and supS2.
4. LetS⊆Dbe a dense subset ofDandf :D→Rbe a continuous function. Suppose that f satisfies the property: for every >0 there existsδ =δ()>0 such that if “ x, y ∈S ” and |x−y|< δ, then
|f(x)−f(y)|< . Prove that f is uniformly continuous on D.
5. (a) Determine whether the function f(x) = sin 1 x
is uniformly continuous on (0,1].
(b) Determine whether the function f(x) =
(
xsin 1 x
, 0< x≤1
0, x= 0
is uniformly continuous on [0,1].
6. Use the Bolzano - Weierstrass Theorem to prove the Extreme Value Theorem.
7. Let f :D⊆R →Rn be a vector valued function. We say that f is uniformly continuous if for every >0 there exists δ =δ()>0 such that
kf(x)−f(y)k< whenever |x−y|< δ.
Denotef = f1,· · · , fn
. Prove thatf is uniformly continuous if and only iffi is uniformly continuous for i= 1,· · · , n.
Lecture Note:
(Page 47)
1. Problem 1.11 2. Problem 1.13 3. Problem 1.14 4. Problem 1.15
5. Problem 1.23(3)(5)(9)
