Post on 31-Dec-2015
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MAT 1234Calculus I
Section 1.8
Continuity
http://myhome.spu.edu/lauw
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Continuous at a point Continuous on an open interval How to demonstrate that a function is not
continuous Intermediate Value Theorem
Preview
Polynomials are “nice” functions when computing limits
Let us look into this “nice” property Other “nice” functions
1. Continuous at a Point
Definition
A function f is continuous at a point a if
)()(lim afxfax
Definition
A function f is continuous at a point a if
)()(lim afxfax
That means…
1.
2.
3.
Example 0 (Section 1.5)
5.1)(lim2
xfx
5.1)(lim2
xfx
5.1)(lim2
xfx
x
y
2
1.5
y=f(x)
Example 0 (Section 1.5)
5.1)(lim2
xfx
5.1)(lim2
xfx
5.1)(lim2
xfx
x
y
2
1.5
y=f(x)
5.1)2( f
Example 1
Explain why the fun. is discontinuous at x=1
1 if 2
1 if 1
1)(
x
xxxf
Note that we need to explain this theoretically. The diagram only serves as an aid.
Example 2
Explain why the fun. is discontinuous at x=1
1 if 4
1 if 1)(
2
xx
xxxf
2. Continuous on an Open Interval
Definition
A function f is continuous on an open interval if it is continuous at every number of the interval.
What if …
If the interval is not open, the definition above breaks down at the end points.
What if …
If the interval is not open, the definition above breaks down at the end points.
A different (modified) definition is required.
We will skip that. Read the text if you are interested.
Common Continuous Functions
Polynomials Rational Functions Root Functions Tri. Functions
Continuous at every no. in their domains
Combinations of Continuous Functions
If f and g is continuous at a, then f+g, f-g, fg, f/g*, cf are also continuous at a.
(*g(a)≠0) ( )y f x
( )y g x
( ) ( )y f x g x
Combinations of Continuous Functions
If g is continuous at a, and f is continuous at g(a), then the composite function
is also continuous at a.f g
Combinations of Continuous Functions
If g is continuous at a, and f is continuous at g(a), then the composite function
is also continuous at a.
In practice, find the composite function and determine its continuity.
f g
Example 32Let ( ) 1 and ( ) .
Is ( ) continuous at 0?
g x x f x x
f g x x
Example 4
For what value of c is the function f continuous at ?
2
1 if 2( )
1 if 2
cx xf x
cx x
2x
Example 4
For what value of c is the function f continuous at ?
2
1 if 2( )
1 if 2
cx xf x
cx x
2x
2
2
(2)
lim ( )
lim ( )
is continuous at 2 if
x
x
f
f x
f x
f x
Expectations2
1 if 2( )
1 if 2
cx xf x
cx x
2
2
(2)
lim ( )
lim ( )
is continuous at 2 if
x
x
f
f x
f x
f x
3. Intermediate Value Theorem
Intermediate Value Theorem
Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b)
Then there is a no. c in (a,b) such that
f(c)=N
Intermediate Value TheoremSuppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b)
x
y
b
y=f(x)
a
f(a)
f(b)
N
Intermediate Value TheoremApplications
Use to prove other theorems Use to estimate the roots of equations
• Find a such that f(a)=0
Intermediate Value TheoremApplications
x
y
b
y=f(x)
af(a)
f(b)
c
0
Suppose f(a) and f(b) are with different signs
Then there is a no. c in (a,b) such that f(c)=0
Review: We learned…
Continuous at a point Continuous on an open interval Intermediate Value Theorem Note that in the exam, you may be ask to
state the theorems you learned in this class.