Exercise # 3
MA4542 Real Analysis
1. (a) Show that f (A ) = f (A ).
(b) Show that f (A ) f (A ).
(c) Give an example where f (A ) 6= f (A ).
2. (a) Show that if f maps X into Y and A X, B Y , then
f f 1 (B) B and f 1 (f (A) A.
(b) Give examples to show th

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MA4542
Exercise # 2 Solutions1
1. Let f be a real-valued function defined on R. Show that the set of points at
which f is continuous is a G set.
Proof. Define
w(x) := lim supcfw_|f (x1 ) f (x2 )| : x1 , x2 (x , x + ).
0
We next show that (i) Et := cfw_x R

MA4542
Midterm Exam
24/10/2016
ID:
Name:
Signature:
a) Total Number of Questions: 4.
b) Try to answer all questions.
c) Show your steps.
(Q1) Show that the following statements are equivalent.
(i) E is measurable.
(ii) For any > 0, there is a open set O E

MA4542
Exercise # 2
1. Let f be a real-valued function defined on R. Show that the set of points at
which f is continuous is a G set.
2. Let f : R R be a continous function defined on R. Show that the inverse
image of a Borel set under f is Borel.
3. Show

MA4542 Real Analysis
Exercise # 1
1. (a) Show that f (A ) = f (A ).
(b) Show that f (A ) f (A ).
(c) Give an example where f (A ) 6= f (A ).
2. (a) Show that if f maps X into Y and A X, B Y , then
f f 1 (B) B and f 1 (f (A) A.
(b) Give examples to show th

MA4542
Solution to Assignment # 1
1. Solution:
(a) If y f (A ), then there exists some x A such that y = f (x). That
means x A for some . It follows that y = f (x) f (A ) f (A ). This proves
the inclusion f (A ) f (A ).
If y f (A ), then y f (A ) for some

MA4542
Take Home Assignment #1
Due on Monday, October 24, 20161
1. Show that a strictly increasing function that is defined on an interval has a
continuous inverse.
2. Let X be a set, and let Y X be a subset. For any F P(X), let FY :=
cfw_F Y : F F.
(i) L

MA4542 Basic Exercises1
1. Prove the De Morgans identities: ( E )c = Ec and ( E )c = Ec .
2. Show that (i) f 1 (E c ) = (f 1 (E)c ; (ii) f 1 ( E ) = f 1 (E ); (iii) f 1 ( E ) =
f 1 (E ). (iv) f 1 (E1 \E2 ) = f 1 (E1 )\f 1 (E2 ).
3. State the definitions

MA4542
Take Home Assignment #1
Due on Wednesday, October 16, 2013
1. Show that a strictly increasing function that is defined on an interval has a
continuous inverse.
Proof. Let f : I E be a strictly increasing function with I being an interval
and E = f

MA4542
Take Home Assignment #21
1. (a) Let f be a bounded measurable function on E. Show that there are sequences of simple functions on E, cfw_n and cfw_n , such that cfw_n is increasing
and cfw_n is decreasing and each of these sequences converges to