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 inte
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 (
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
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
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 , th
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 measur
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
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 nu
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_x
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 :
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