Continuity and One-Sided Limits

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Continuity and One-Sided Limits

• Lesson 2.4

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Don't let this Happen to you!
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Intuitive Look at Continuity

• A function withoutbreaks orjumps
• The graph can bedrawn without lifting the pencil

?
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Continuity at a Point

• A function can be discontinuous at a point
• A hole in the function and the function not
  defined at that point
• A hole in the function, but the function is
  defined at that point

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Continuity at a Point

• A function can be discontinuous at a point
• The function jumps to a different value at a
  point
• The function goes to infinity at one or both
  sides of the point, known as a pole

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Definition of Continuity at a Point

• A function is continuous at a point x c if the
  following three conditions are met
• f(c) is defined
• For this link, determine which of the conditions
  is violated in the examples of discontinuity

x c
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"Removing" the Discontinuity

• A discontinuity at c is called removable if
• If the function can be made continuous by
• defining the function at x c
• or redefining the function at x c
• Go back to this link, determine which (if any) of
  the discontinuities can be removed

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Which of These is Dis/Continuous?

• When x 1 why or not

Are any removable?
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Continuity Theorem

• A function will be continuous at any number x
  c for which f(c) is defined, when
• f(x) is a polynomial
• f(x) is a power function
• f(x) is a rational function
• f(x) is a trigonometric function
• f(x) is an inverse trigonometric function

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Properties of Continuous Functions

• If f and g are functions, continuous at x
  cThen
• is continuous (where s is a
  constant)
• f(x) g(x) is continuous
• is continuous
• is continuous
• f(g(x)) is continuous

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One Sided Continuity

• A function is continuous from the right at a
  point x a if and only if
• A function is continuous from the left at a point
  x b if and only if

a
b
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Continuity on an Interval

• The function f is said to be continuous on an
  open interval (a, b) if
• It is continuous at each number/point of the
  interval
• It is said to be continuous on a closed interval
  a, b if
• It is continuous at each number/point of the
  interval and
• It is continuous from the right at a and
  continuous from the left at b

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Continuity on an Interval

• On what intervals are the following functions
  continuous?

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Intermediate Value Theorem

• Given function f(x)
• Continuous on closed interval a, b
• And L is a number strictly between f(a) and f(b)
• Then there exists at least one number c on the
  open interval (a, b) such that f(c) L

f(b)
L
f(a)
b
a
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Locating Roots with Intermediate Value Theorem

• Given f(a) and f(b) have opposite sign
• One negative, the other positive
• Then there must be a root between a and b

a
Try exercises 88, 90, and 92 pg 100
b
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Assignment

• Lesson 2.4A
• Page 98
• Exercises 1 49 odd
• Lesson 2.4B
• Page 99
• Exercises 51 75, 85 105 odd