Download Math 3100H – Homework #8 posted April 22, 2016

Math 3100H – Homework #8
posted April 22, 2016; due at the start of class on May 2, 2016
Assignments are expected to be stapled and neatly written; illegible work may not be graded.
Problems:
1. If A is a subset of real numbers, an element a ∈ A is called an isolated point of A
if a ∈
/ A0 . For example, if A = {0, 1, 1/2, 1/3, . . . }, then every nonzero element of A is
isolated, while if A = [0, 1] ∪ {2}, then 2 is the only isolated point of A.
Show that an element a ∈ A is an isolated point if and only if there is some > 0 with
(a − , a + ) ∩ A = {a}.
2. Let A be a nonempty subset of the real numbers that is bounded above. Let U =
lub(A). Assuming U ∈
/ A, show that U ∈ A0 .
3. Use the limit definition (Definition 4.2.1 in Abbott’s book) to prove the following
statements.
(a) limx→2 (3x − 5) = 1
(b) limx→0 x3 = 0
(c) limx→2 x3 = 8
x
does not exist.
x→0 |x|
4. Prove that lim
5. Let D(x) be the function defined for every real number x by
(
1 if x is rational,
D(x) =
0 if x is irrational.
(This is called Dirichlet’s function.) At which real numbers c is D continuous?
Justify your answer.
Hint: We proved in class that every open interval contains a rational number. As
a lemma for this problem, show√that every open interval also contains an irrational
number. You may assume that 2 is irrational.
6. In this exercise, we establish the Limit Criterion for Continuity discussed in class. Let
A be a subset of the real numbers.
(a) Suppose that c ∈ A and c is not a limit point of A. Give a careful proof that
every function f : A → R is continuous at c.
Hint: We did a specific example of this in class. To handle the general case, use
the result of Problem #1.
(b) Suppose that c ∈ A and c is a limit point of a. Show that f is continuous at c if
and only if limx→c f (x) = f (c).
7. Using the definition of continuity, show that constant functions are continuous on
A = R, and that the function f (x) = x is continuous on A = R.
8. Suppose that h : R → R is continuous for all real numbers.
(a) Let Z = {c ∈ R : h(c) = 0}. Show that every limit point of Z belongs to Z.
Hint: Use the sequential criterion for limit points and the sequential criterion for
continuity.
(b) Show that if h(c) = 0 for every rational number c, then h(c) = 0 for all real
numbers c.
Hint: Recall that every real number is a limit point of the set of rational numbers.
Now argue as in (a).
9. Recall that every rational number can be written uniquely in the form p/q, where p, q
are integers, q is positive, and p and q have no common factor > 1. (This is called the
lowest terms representation of the rational number.) Let T (x) be the function defined
for every real number x by the rule
(
1/q if x = p/q in lowest terms,
T (x) =
0
if x is irrational.
(This is Thomae’s function.) Prove that if c is irrational, then T is continuous at c,
but that if c is rational, then T is not continuous at c.
Hint: See p. 102 of Abbott’s book for the method to prove T is discontinuous at
rational points.
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