Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
11/11/2008
Uniform Continuity

Definition 4.3.1. Let E be a subset of R and f a
real-valued function with domain E, that is,
f : ER. The function f is uniformly continuous on
E if   0,   0, such that
x, y  E with | x  y |  | f ( x)  f ( y ) | 

Remark. The choice of δ depends only on ε.
Question: how would you define that f is not
uniformly continuous on E?
Examples for discussion
1. f ( x)  x on [a, b]
2
2. f ( x)  x on [0, )
2
3. f ( x)  sin( x) on (, )
1
4. f ( x)  on (0, )
x
Lipschitz Functions

Function f : ER satisfies a Lipschitz condition on
E if
M  0 such that
| f ( x)  f ( y ) | M | x  y | (x, y  E ).

Functions satisfy the Lipschitz condition are called
Lipschitz functions.
Theorem on Lipschitz
Functions


Theorem 4.3.3 Suppose f : ER is a Lipschitz
function on E, then it is uniformly continuous on E.
Example.
1.
f ( x)  sin( x) on (, ).
Is it a Lipschitz function ?
2.
f ( x)  x
on (0, ).
Is it a Lipschitz function ?
Uniform Continuity Theorem

Theorem 4.3.4. If K is a compact subset of R and
f: KR is continuous on K, then f is uniformly
continuous on K.

Corollary 4.3.5. A continuous real-valued function
on a closed and bounded interval [a, b] is uniformly
continuous.
The Derivative

Definition 5.1.1 Let I be an interval and let f : IR. For fixed
p є I, the derivative of f at p, denoted f’(p), is defined to be
f ' ( p)  lim
x p
or
 lim
h 0
f ( x)  f ( p )
x p
f ( p  h)  f ( p )
h
provided the limit exists. If f’(p) is definded at a point p є I,
we say that f is differentiable at p. If the derivative f’ is
defined at every point of a set E, we say that f is
differentiable on E.
The Right Derivative

Definition 5.1.2 Let I be an interval and let f : IR.
For fixed point p є I (p is not the right end point of
I), the right derivative of f at p, denoted
f ' ( p ), is defined as
f ' ( p )  lim
x p 
f ( x)  f ( p)
x p
f ( p  h)  f ( p )
 hlim
0 
h
or
provided the limit exists.
The Left Derivative

Definition 5.1.2 Let I be an interval and let f : IR.
For fixed point p є I (p is not the left end point of I),
the left derivative of f at p, denoted
f ' ( p ), is defined as
f ( x)  f ( p)
f ( p )  lim
x p 
x p
'

f ( p  h)  f ( p )
 hlim
0 
h
or
provided the limit exists.
The Derivative

Remark. Let I be an interval and let f : IR. For fixed point
p є int(I), the derivative of f at p exists if and only if
f ( p)  f ( p).
'

'


Theorem 5.1.4. If I is an interval and f:IR is differentiable
at p є I, then f is continuous at p.

Remark: The converse of the theorem is not true.
Derivatives of Sums, Products,
and Quotients

Theorem 5.1.5 Suppose f ,g are real-valued
functions defined on an interval I. If f and g are
differentiable at x є Int(I), then f+g, fg, and f/g (if
g(x)≠0) are differentiable at x and
(a) ( f  g )' ( x)  f ' ( x)  g ' ( x),
(b) ( fg )' ( x)  f ' ( x) g ( x)  f ( x) g ' ( x)
'
f
f ' ( x) g ( x)  f ( x) g ' ( x)
(c)   ( x) 
, provided g ( x)  0.
2
( g ( x))
g
The Chain Rule

Theorem 5.1.6 Suppose f is a real-valued function
defined on an interval I and g is a real-valued
function defined on some interval J such that
Range f is a subset of J. If f is differentiable at x є
Int(I) and g is differentiable at f(x), then g◦f is
differentiable at x and
( g  f )' ( x)  g ' ( f ( x)) f ' ( x).