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 in a casual definition if you can draw
it in one motion without picking up your pencil
a more thorough definition
A function is continuous at a point if the limit
is the same as the value of the function.
lim f(x) f(c) lim f(x) x?c-
x?c
The function at the left has discontinuities at
x1 and x2.
f(x) is continuous at x0 and x4, because the
one-sided limit matches the function value at the
endpoint.
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Removable Discontinuity
(You could fill the gap and establish
continuity by adding a single point to the
function.)
Essential Discontinuities
oscillating
infinite
jump
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Removing a point discontinuity
To write an extended function that is continuous
at x 1, determine the limit of f(x)
and introduce the missing condition (from
the limit at x1) to create the extended,
continuous function. (Note The essential
discontinuity at x -1 can not be
removed!)
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Removing a discontinuity from a graphical
perspective

(Also note the discontinuity at x -1 that
cannot be removed.)
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Continuous functions can be added, subtracted,
multiplied, divided, multiplied by a constant, or
composed, and the new function remains
continuous.
examples
Outer function y sin(x)
Outer function y abs(x)
Inner function y x2
Inner function y cos(x)
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Intermediate Value Theorem
If a function is continuous between a and b, then
it takes on every function value between
and
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Example 5
Is any real number exactly one less than its cube?
f(1) -1 and
f(2) 5
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MATH
PRB
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int(
The calculator connects the dots which covers
up the discontinuities.
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Graphing calculators can make non-continuous
functions appear continuous.
If we change the resolution to 1, then we get a
graph that is closer to the correct floor graph.
The open and closed endpoints do not show, but
we can see the discontinuities more clearly!
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