MAT 1234 Calculus I Section 1.8 Continuity .
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Transcript of MAT 1234 Calculus I Section 1.8 Continuity .
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MAT 1234Calculus I
Section 1.8
Continuity
http://myhome.spu.edu/lauw
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Preview
Continuous at a point Continuous on an open interval How to demonstrate that a function is not
continuous Intermediate Value Theorem
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Preview
Polynomials are “nice” functions when computing limits
Let us look into this “nice” property Other “nice” functions
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1. Continuous at a Point
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Definition
A function f is continuous at a point a if
)()(lim afxfax
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Definition
A function f is continuous at a point a if
)()(lim afxfax
That means…
1.
2.
3.
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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)
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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
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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.
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Example 2
Explain why the fun. is discontinuous at x=1
1 if 4
1 if 1)(
2
xx
xxxf
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2. Continuous on an Open Interval
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Definition
A function f is continuous on an open interval if it is continuous at every number of the interval.
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What if …
If the interval is not open, the definition above breaks down at the end points.
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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.
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Common Continuous Functions
Polynomials Rational Functions Root Functions Tri. Functions
Continuous at every no. in their domains
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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
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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
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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
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Example 32Let ( ) 1 and ( ) .
Is ( ) continuous at 0?
g x x f x x
f g x x
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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
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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
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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
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3. Intermediate Value Theorem
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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
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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
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Intermediate Value TheoremApplications
Use to prove other theorems Use to estimate the roots of equations
• Find a such that f(a)=0
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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
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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.