1003 Continuity
General Idea: ______________________________________________________________________________
We already know the continuity of many functions:
Polynomial (Power),
Rational,
Radical,
Exponential, Trigonometric, and Logarithmic functions
DEFN: A function is continuous on an interval if it is continuous at each point in the interval.
DEFN: A function is continuous at a point c IFF
a) __________________________________________
_____________________________________
b) __________________________________________
_____________________________________
c) __________________________________________
_____________________________________
THEOREM:
Note: All three steps are implied.
One sided Continuity:
A function y
= f(x) is continuous at a , the left endpoint a of its domain if lim f ( x) f (a ) .
A function y
= f(x) is continuous at b , the right endpoint a of its domain if lim f ( x) f (b) .
x a
xb
Theorem:
A function is Continuous on a closed interval if it is continuous at every point in the open interval and
continuous from one side at the end points.
Example :
The graph over the closed interval [-2,4] is given.
Discontinuity
Continuity may be disrupted by:
(a).
c
c
(b).
c
(c).
c
Removable or Essential ( Non-Removable) Discontinuities
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__________________________________________________________________________________________
Examples:
Identify the x-values (if any) at which f(x) is not continuous.
Identify the reason for the discontinuity and the type of discontinuity.
Is the discontinuity removable or essential?
f ( x)
x 2
x4
f ( x)
1
( x 3) 2
3 x, x 1
f ( x)
3 x, x 1
Identify the x-values (if any) at which f(x) is not continuous.
Identify the reason for the discontinuity and the type of discontinuity.
Is the discontinuity removable or essential?
Algebraic Method
3x 2 x 2
f ( x) 2
3x 4 x 2
Reason
Type
x = -3
____________________
________________
x = -2
____________________
________________
x=0
____________________
________________
x=1
____________________
________________
x=2
____________________
________________
x=3
____________________
________________
1- x 2
x 1
2
f ( x) x - 2 1 x 3
x2 9
x3
x 3
Find c that makes f ( x) continuous.
4 x 2c x 5
f ( x) 2
x c x 5
Consequences of Continuity:
A. INTERMEDIATE VALUE THEOREM
** Existence Theorem
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_______________________________________________
_______________________________________________
c
_______________________________________________
EX: Verify the I.V.T. for f (c).
Then find c.
f ( x) x 2
on
f (c) 3
[1, 2]
EX. Zero Locater Corollary.
Show that the function f ( x) x3 2 x 1 has a ZERO on the interval [0,1].
( CALCULUS AND THE CALCULATOR:
The calculator looks for a SIGN CHANGE
between Left Bound and Right Bound.)
EX. Sign on an Interval - Corollary
(Number Line Analysis)
1
3
x 1 2
( x 1)( x 2)( x 4) 0
B. EXTREME VALUE THEOREM
On every closed interval there exists an absolute maximum value and minimum value
y
x
.
Assignment 1003:
FDWK p. 84
11)
a) does f(-1) exist?
b) does lim f ( x) exist?
x1
c) does lim f ( x) f ( 1) ?
x1
d) Is f(x) continuous a t f(-1)?
12)
a) does f(1) exist?
b) does lim f ( x) exist?
x1
c) does lim f ( x) f (1) ?
x1
d) Is f(x) continuous a t f(1)?
13) Is f defined at x = 2 ? Is f continuous at x = 2?
14) At what values of x is f continuous?
15) What value should be assigned to f(2) to make the extended function continuous at x = 2 .
16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 .
17) Is it possible to extend f to make it continuous at x = 0 ? Explain.
18) Is it possible to extend f to make it continuous at x = 3 ? Explain.
(a) Find each point of discontinuity. Give the reason for your answer. (b) Which of the discontinuities are
removable? essential?
19)
3 x , x 2
f ( x) x
2 1 , x 2
20)
47) Find a value for a so that the function
is continuous.
x2 1 , x 3
f ( x)
2ax , x 3
49) Find a value for a so that the function
is continuous.
4 x 2 , x 1
f ( x) 2
ax -1 , x 1
3 x , x 2
f ( x) 2 ,
x2
x
, x2
2
21)
1
, x 1
f ( x) x 1
x3 2 x 5 , x 1
48) Find a value for a so that the function.
is continuous
2x 3 , x 2
f ( x)
ax 1 , x 2
50) Find a value for a so that the function.
is continuous
x2 x a , x 1
f ( x) 3
, x 1
x
0
