Calculus 2.3

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2.3 Continuity
Grand Canyon, Arizona
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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 x1 and x2.
It is continuous at x0 and x4, because the
one-sided limits match the value of the function
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Removable Discontinuities
(You can fill the hole.)
Essential Discontinuities
oscillating
infinite
jump
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Removing a discontinuity
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Removing a discontinuity
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Continuous functions can be added, subtracted,
multiplied, divided and multiplied by a constant,
and the new function remains continuous.
examples
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Intermediate Value Theorem
If a function is continuous between a and b, then
it takes on every value between and
.
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Example 5
Is any real number exactly one less than its cube?
(Note that this doesnt ask what the number is,
only if it exists.)
1 solve
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Graphing calculators can sometimes make
non-continuous functions appear continuous.
CATALOG
F
floor(
This example was graphed on the classic TI-89.
You can not change the resolution on the Titanium
Edition.
The calculator connects the dots which covers
up the discontinuities.
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Graphing calculators can make non-continuous
functions appear continuous.
CATALOG
F
floor(
If we change the plot style to dot and the
resolution to 1, then we get a graph that is
closer to the correct floor graph.
The open and closed circles do not show, but we
can see the discontinuities.
p