Continuity and OneSided Limits (1.4)
September 26th, 2012
I. Continuity at a Point on an Open
Interval
Def: A function is continuous at a point c if
1. f(c) is defined
lim f (x)
2.
exists, and
xc
lim f (x)  f (c) .
3.
xc
A function is continuous on an open interval (a, b)
if it is continuous at each point in the interval. A
function that is continuous on the entire real line
is everywhere continuous.
(,)
If a function f is continuous on the open interval
(a, b) except at point c, it is said to have a
discontinuity at c. This discontinuity is removable
if f can be made continuous by appropriately
defining (or redefining) f(c). Otherwise, the
discontinuity is nonremovable.
f(c) is not
defined
Removable
discontinuity
at c.
lim f (x)
xc
does not exist
Nonremovable
discontinuity
at c.
lim f (x)  f (c)
xc
Removable
discontinuity at
c.
Ex. 1: Discuss the continuity of each function.
2
f (x) 
(a)
x5
(b)
x 2  2x 1
g(x) 
x 1
(c)
 x 2  3, x  0
h(x)  
2x  3, x  0
(d)
k(x)  2 cos 3x
II. One-Sided Limits and Continuity on
a Closed Interval
One-Sided Limits are denoted by the following.
means the limit as x
lim f (x)
approaches
xc
lim f (x)
xc
c from the right, and
means the limit as x
approaches
c from the left.
Ex. 2: Find the limit (if it exists). If it does not
exist, explain why.
x
(a) lim
x3
(b)
x 9
2
lim [[x  2]] 1
x1
(c)
lim [[x  2]] 1
x1
You try:
(a)
x5
lim 2
x5 x  25
(b)
x
lim
x0 x
Thm. 1.10: The Existence of a Limit: Let f be a
function and let c and L be real numbers. The
limit of f(x) as x approaches c is L if and only if
lim f (x)  L
xc
and
lim f (x)  L
xc
Def: A function f is continuous on a closed interval
[a, b] if it is continuous on the open interval (a, b)
and
and lim f (x)  f (b)
lim f (x)  f (a)
xb
xa
.
We say that f is continuous from the right of a and
continuous from the left of b.


Ex. 3: Discuss the continuity of the function
1
g(x)  2
on the closed interval [-1, 2].
x 4
III. Properties of Continuity
Thm. 1.11: Properties of Continuity: If b is a real
number and f and g are continuous at x=c, then
the following functions are also continuous at c.
1. Scalar multiple: bf
2. Sum and difference: f  g
3. Product: fg
f
, g(c)  0
4. Quotient:
g
Functions that are Continuous at Every Point in
their Domain:
n
n1
p(x)

a
x

a
x
 ... a1x  a0
n
n1
1. Polynomial functions
2. Rational functions
3. Radical functions
p(x)
r(x) 
,q(x)  0
q(x)
f (x)  n x
4. Trigonometric functions
sin x,cos x, tan x,csc x,sec x,cot x
Thm. 1.12: Continuity of a Composite Function: If
g is continuous at c and f is continuous at g(c),
( f og)(x)
then the composite function given
by  f (g(x))
is continuous at c.
Ex. 4: Find the x-values (if any) at which f is not
continuous. Which of the discontinuities are
removable?
(a)
f (x)  3x  cos x
2
(b)
 1
 1
f (x)  2    4    3
 x
 x
IV. The Intermediate Value Theorem
***Thm. 1.13: The Intermediate Value Theorem: If
f is continuous on the closed interval [a, b] and k
is any number between f(a) and f(b), then there is
at least one number c in [a, b] such that f(c)=k.
Ex. 5: Explain why the function f (x)  x  2  cos x
has a zero in the interval 0,  
.
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