Advanced Calculus Assignment 2

1. Advanced Calculus, Assignment 2
1. Let f : R → R be defined by
f(x) =
x if x is rational
x2
if x is irrational
.
Using the definition of continuity directly prove that f is continuous at 1
and discontinuous at 2.
2. Let f : R → R and g : R → R be continuous functions. Suppose that for
all rational numbers r, f(r) = g(r). Show that f(x) = g(x) for all x ∈ R.
3. Let f : A → R be a function and c ∈ A such that for every sequence
{xn}n∈N in A with limn→∞ xn = c, the sequence {f(xn)}n∈N converges.
Show that f is continuous at c.
4. Let I = [a, b] be a closed bounded interval and f : I → R be continuous
on I.
(a) Show that if k ∈ R is a number satisfying
inf f(I) ≤ k ≤ sup f(I)
then there is a number c ∈ I such that f(c) = k.
(b) Show that the set f(I) is a closed bounded interval or a single number.
5. (a) Show that the polynomial p(x) = x3
− 2x2
+ x − 1 has a real root in
(1, 2).
(b) Suppose that f is a polynomial of odd degree d.
i. Write f(x) = adxd
+ . . . + a1x + a0 where ad = 0. Divide f by
ad to obtain
g(x) = xd
+ . . . +
a1
ad
x +
a0
ad
.
Prove that g(n) is positive for some large n ∈ N, and g(−k) is
negative for some large k ∈ N.
ii. Show that f has a root in R.
6. (a) Find an example of a discontinuous function f : [0, 1] → R where the
Intermediate Value Theorem fails.
(b) Let
f(x) =
sin 1
x if x = 0
0 if x = 0
.
Show that f has the Intermediate Value Property. That is, for any
a < b, if there exists a y such that f(a) < y < f(b) or f(b) < y <
f(a), then there exists a c ∈ (a, b) such that f(c) = y.
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2. 7. Suppose f : [0, 1] → [0, 1] is a continuous function. Show that f has a
fixed point, in other words, show that there exists an c ∈ [0, 1] such that
f(x) = x.
8. (a) A function f : R → R is said to be periodic on R if there exists
a number p > 0 such that f(x + p) = f(x) for all x ∈ R. Prove
that a continuous periodic function on R is bounded and uniformly
continuous.
(b) Prove that if f : A → R is uniformly continuous, then for any B ⊂ A,
f|B is also uniformly continuous.
9. (a) Prove that g : [0, ∞) → R : x →
√
x is uniformly continuous.
(b) Let f : (0, 1) → R be a bounded continuous function. Show that the
function g(x) : x(1 − x)f(x) is uniformly continuous.
10. Let A, B be intervals in R. Let f : A → R and g : B → R be uniformly
continuous functions such that f(x) = g(x) for all x ∈ A ∩ B. Define the
function h : A ∪ B → R by
h(x) =
f(x) if x ∈ A
g(x) if x ∈ B A
(a) Prove that if A ∩ B = ∅, then h is uniformly continuous.
(b) Find an example where A ∩ B = ∅ and h is not even continuous.
11. Let f(x) = sin(πx) on R. Show that limx→∞ f(x) does not exist.
12. Let f : [1, ∞) → R be a function. Define g : (0, 1] → R via g(x) = f 1
x .
Show that limx→0+ g(x) exists if and only if limx→∞ f(x) exists, in which
case these limits are equal.
13. Suppose f : [0, 1] → R is a monotone function. Prove that f is bounded.
14. Show that if I = [a, b] and f : I → R is increasing on I, then f is
continuous at a if and only if f(a) = inf{f(x) : x ∈ (a, b]}.
15. Let {qn}∞
n=1 be an enumeration of the set of rational numbers in [0, 1].
Define the function f on [0, 1] by the formula
f(x) =
n: qn<x
1
2n
(a) Prove that f is increasing.
(b) Prove that f is discontinuous at each rational number qm.
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