Sec 2.5: Continuity

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Sec 2.5 Continuity
Continuous Function
Intuitively, any function whose graph can be
sketched over its domain in one continuous motion
without lifting the pencil is an example of a
continuous function.
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Sec 2.5 Continuity
Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example study the continuity at x -1
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Sec 2.5 Continuity
Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example study the continuity at x 4
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Sec 2.5 Continuity
Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example study the continuity at x 2
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Sec 2.5 Continuity
Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example study the continuity at x -2
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Sec 2.5 Continuity
Cont a
Continuity at a Point (end point)
A function f(x) is continues at an end point
a if
Cont from left at a
Cont from right at a
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Sec 2.5 Continuity
Types of Discontinuities.
Which conditions
removable discontinuity
infinite discontinuity
Later oscillating discontinuity
jump discontinuity
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Sec 2.5 Continuity
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Sec 2.5 Continuity
Exam1-102
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Sec 2.5 Continuity
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Sec 2.5 Continuity
Exam1-122
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Sec 2.5 Continuity
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Sec 2.5 Continuity
The inverse function of any continuous one-to-one
function is also continuous.
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Sec 2.5 Continuity
Inverse Functions and Continuity
The inverse function of any continuous one-to-one
function is also continuous.
This result is suggested from the observation
that the graph of the inverse, being the
reflection of the graph of ƒ across the line y x
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Sec 2.5 Continuity
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Sec 2.5 Continuity
continuous
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Sec 2.5 Continuity
Exam1-101
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Sec 2.5 Continuity
Geometrically, IVT says that any horizontal line
between ƒ(a) and ƒ(b) will cross the curve at
least once over the interval a, b.
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Sec 2.5 Continuity
The Intermediate Value Theorem
y0ƒ(c)
1) ƒ(x) cont on a,b
2) y0 between ƒ(a) and ƒ(b)
c in a,b
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Sec 2.5 Continuity
One use of the Intermediate Value Theorem is in
locating roots of equations as in the following
example.
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Sec 2.5 Continuity
E1 TERM-121
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Sec 2.5 Continuity
Exam1-101
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Sec 2.5 Continuity