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1-4: Continuity and One-Sided LimitsObjectives:•Define and explore properties of continuity•Discuss one-sided limits•Introduce Intermediate Value Theorem
©2002 Roy L. Gover (roygover@att.net)
Definitionf(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.
ExamplesDiscontinuous FunctionsRemovable discontinuityJump Discontinuity (non-
removable)Infinite discontinuity (non-removable)
Definition
f(x) is continuous at x=c if and only if:1. f (c) is defined …and
lim ( )x c
f x
2. exists …and
lim ( ) ( )x c
f x f c
3.
Definition
f(x) is continuous on the open interval (a,b) if and only if f(x) is continuous at every point in the interval.
Try ThisFind the values of x (if any) where f is not continuous. Is the discontinuity removable?
2
0, for 0
, for 0
x
x x
Continuous for all x
( )f x
Try ThisFind the values of x (if any) where f is not continuous. Is the discontinuity removable?1
( )f xx
Discontinuous at x=o, not removable
Definition
f(x) is continuous on the closed interval [a,b] iff it is continuous on (a,b) and continuous from the right at a and continuous from the left at b.
Example
a
b
f(x)
f(x) is continuous on (a,b)
f(x) is continuous from the right at a
f(x) is continuous from the left at b
f(x) is continuous on [a,b]
Try This
Use the graph to determine the limit, the limit from the right & the limit from the left as x0.
Try This
Use the graph to determine the limit, the limit from the right & the limit from the left as x1.
x=1
Intermediate Value Theorem
Theorem 1.13: If f is continuous on [a,b] and k is a number between f(a) & f(b), then there exists a number c between a & b such that f(c ) =k.
Intermediate Value Theorem•an existence theorem; it
guarantees a number exists but doesn’t give a method for finding the number.•it says that a continuous function never takes on 2 values without taking on all the values between.
ExampleRyan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.
Try ThisUse the Intermediate Value Theorem to show that 3( )f x x
has a zero in the interval [-1,1].
Solution3( )f x x
( 1) 1
(1) 1
f
f
therefore, by the Intermediate Value Theorem, there must be a f (c)=0 where
1 1c
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