Section 1.4 ‐ Continuity and
One‐Sided Limits
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Continuity
Informally ­ You can graph the function without lifting up your pencil.
A function is continuous at x=c, if it is unbroken at c. Title: Aug 28 ­ 8:23 AM (2 of 9)
Continuity
Formal Definition­ f is continuous at c, if:
1. f(c) is defined
lim f(x) 2. exists.
x c
3. lim f(x) = f(c) x c
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All 3 Conditions must be satisfied!!!
Using the graphs below, determine which condition of continuity is not met.
1.
2.
c
c
3. c
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removable and non­removable
discontinuities
Removable
The function can be made continuous by redefining f(c).
We can "fill in the dot" by defining
f(2) = 4
Or...We can redefine the function as f(x) = x+2
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Nonremovable
There is no way to define f(c) to make this continuous.
Discontinuity at c = 1
one­sided limits
Limit from the left
Limit from the right
Must be equal for the
function to be
continuous.
Example 1
Not Continuous
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Example 2
Not Continuous
intermediate value theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b),
then there is at least one number c in [a,b] that f(c) = k .
Pick a value of k between f(a) and f(b).
Can you find a c?
f(b)
a
b
f(a)
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intermediate value theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b),
then there is at least one number c in [a,b] that f(c) = k .
What happens when the function is not continuous?
Pick a value of k between f(a) and f(b).
Can you find a c for every k?
f(b)
a
b
f(a)
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Homework:
p76: 1­17 eoo, 25, 28, 33 ­ 57 eoo, 66, 77, 81, 83, 87, 90, 100, 101
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