1003 Continuity

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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
xb
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
__________________________________________________________________________________________
__________________________________________________________________________________________
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
x4
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

x3
 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
_______________________________________________
_______________________________________________
_______________________________________________
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?
x1
c) does lim f ( x)  f ( 1) ?

x1
d) Is f(x) continuous a t f(-1)?
12)
a) does f(1) exist?
b) does lim f ( x) exist?
x1
c) does lim f ( x)  f (1) ?
x1
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 ,
x2
x
 , x2
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
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