2.3 continuity. most of the techniques of calculus require that functions be continuous. a function...
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2.3 Continuity
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Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function.
This function has discontinuities at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function
1 2 3 4
1
2
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jump infinite oscillating
Essential Discontinuities:
Removable Discontinuities:
(You can fill the hole.)
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Removing a discontinuity:
3
2
1
1
xf x
x
has a discontinuity at .1x
Write an extended function that is continuous at .1x
3
21
1lim
1x
x
x
2
1
1 1lim 1 1x
x x xx x
1 1 1
2
3
2
3
2
1, 1
13
, 12
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
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Removing a discontinuity:
3
2
1, 1
13
, 12
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
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Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.
Also: Composites of continuous functions are continuous.
examples: 2siny x cosy x
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Intermediate Value Theorem
If a function is continuous between a and b, then it takes
on every value between and . f a f b
a b
f a
f b
Because the function is continuous, it must take on every y value between and .
f a f b
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Example 5: Is any real number exactly one less than its cube?
(Note that this doesn’t ask what the number is, only if it exists.)
3 1x x
30 1x x
3 1f x x x
1 1f 2 5f
Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5.Therefore there must be at least one solution between 1 and 2.
Use your calculator to find an approximate solution.
3solve 1,x x x 1.32472