MATH 2031 Introduction to Real Analysis
April 16, 2013
Tutorial Note 18
Riemann Integral Cont
Improper Integral
In this part, we focus on functions f (x) which are either unbounded or dened on an interval which is not closed
or not bounded.
(I) Denition:
MATH 2031 Introduction to Real Analysis
January 31, 2013
Tutorial Note 14
Dierentiation (cont)
(I) Generalized Mean Value Theorem
If f, g are continuous on [a, b] and dierentiable on (a, b), then x0 (a, b) such that
g (x0 )(f (b) f (a) = f (x0 )(g(b) g(a)
MATH 2031 Introduction to Real Analysis
September 28, 2012
Tutorial Note 3
Functions
(I) Denition:
A function f from a set A to a set B(denoted by f : A B) is an assignment of every a A to exactly ONE b B
which we denote by f (a), the value of f at a. A f
MATH 2031 Introduction to Real Analysis
January 31, 2013
Tutorial Note 12
Continuity
(C.I) Denition:
A function f : S R is continuous at x0 S i xx f (x) = f (x0 ).
lim
0
xS
lim f (x) = f (x0 ) > 0 > 0 x S, |x x0 | < |f (x) f (x0 )| <
xx0
xS
(C.II) Sequen
MATH 2031 Introduction to Real Analysis
September 19, 2012
Tutorial Note 2
Logic
(I) Negation (or ),
(II) De Morgans laws
(p and q)
(p or q)
( p) or ( q)
,
( p) and ( q)
(III) Quantiers for all , there exist
(IV) Conditional statements p q and (p q) =
MATH 2031 Introduction to Real Analysis
April 9, 2013
Tutorial Note 17
Riemann Integral (Cont Proper Integral)
(I) Denition:
> 0, intervals (a1 , b1 ), (a2 , b2 ) such that
(i) A set S R is of measure zero i
.
(bk ak ) <
(ak , bk ) and
S
k=1
k=1
(ii) A
MATH 2031 Introduction to Real Analysis
March 22, 2013
Tutorial Note 16
Riemann Integral
Proper Integral
In this section, we focus on functions f (x) which are bounded on a closed and bounded interval [a, b]. (f is bounded
on [a, b] if there exists K R, K
MATH 2031 Introduction to Real Analysis
April 16, 2013
Tutorial Note 19
Sequences and Series of Functions
(I) Extended real number system
(i) Denition:
[, +] = R cfw_, + is called the extended real number system.
(ii) Ordering in [, +]
Usual ordering in
MATH 2031 Introduction to Real Analysis
February 13, 2013
Tutorial Note 11
Cauchy sequence
(I) Denition:
cfw_xn is a Cauchy sequence i > 0, K N such that n, m K |xn xm | < .
(II) Cauchy Theorem
cfw_xn converges i it is a Cauchy sequence.
The proof consi
MATH 2031 Introduction to Real Analysis
April 16, 2013
Tutorial Note 20
Sequences and Series of Functions (Cont)
(I) Denition (Pointwise Convergence):
Let E be a set. Then a sequence of functions Sn : E R is said to converge pointwise on E to a function
S
MATH 2031 Introduction to Real Analysis
January 31, 2013
Tutorial Note 13
Dierentiation
(I) Denition:
Let S be an interval of positive length.
A function f : S R is dierentiable at x0 S i f (x0 ) = xx
lim
0
xS
f (x) f (x0 )
exists in R. Also, f is
x x0
di