Continuity

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Continuity One-Sided Limits

• Section 1.4

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After this lesson, you will be able to

• determine continuity at a point and continuity on
  an open interval
• determine one-sided limits and continuity on a
  closed interval
• understand and use the Intermediate Value Theorem

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Continuity at a Point
f is continuous at c if the following three
conditions are satisfied
f(c)
1) f(c) is defined
Muy importante! Be able to recite by heart.
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Some examples of functions that are NOT
continuous at c.

• 1) f is not
  continuous at c because

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Some examples of functions that are NOT
continuous at c.

• 2) f is not
  continuous at c because

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Some examples of functions that are NOT
continuous at c.

• 3) f is not
  continuous at c because

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Continuity on an Open Interval
A function is continuous on an open interval (a,
b) if it is continuous at each point in the
interval.
Continuous Function on the open interval (a, b).
A function that is continuous over the set of
real numbers is called everywhere continuous.
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Discontinuity

• A function that is defined on the interval (a, b)
  (except possibly at c), and is not continuous at
  c is said to have a ___________________ at c.

• There are two types of discontinuities
• Removable If f can be made continuous by
  defining or redefining f(c)this type would
  appear as a _________ in the graph.
• Non-removable f cannot be made continuous by
  changing only f(c)...e.g. a vertical asymptote at
  x c.

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Removable Discontinuity
f
f(c)
Removable discontinuity at xc, since f can be
made continuous by just redefining f(c).
c
Discontinuity at x c
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Non-removable Discontinuity
This function has a non-removable discontinuity
at x c. If only f(c) was redefined, the
function would still be discontinuous.
f
f(c)
c
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Continuity

• Consider the graphs of the following functions
  and discuss the continuity of each.

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One-sided Limits
Limit from the right (right-hand limit)
Limit from the left (left-hand limit)
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One-sided Limits
One-sided limits are great for radical
functions.
Example
We cant take the limit of this function as x
approaches 0 from the left side since negative
numbers are not in the domain. We can only take
the right limit.
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One-sided Limits
Example
Graph the function on the calculator. Then,
determine the limit graphically.
What is the domain of this function?
___________________
One-sided limits are also great for functions
with a closed interval as the domain.
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Examples 1 2
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Example 3
is called the ______________
__________________ ______________.
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Continuity on a Closed Interval
A function is continuous on a closed interval if
it is continuous everywhere inside the interval
and has one-sided continuity at the endpoints.
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Examples

• Discuss the continuity of the function on the
  closed interval.

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Properties of Continuity
If k is a real number and f and g are continuous
at x c, then f g, f-g, fg, kf, and f/g
(provided g(c) ? 0) are also continuous.
The following types of functions are continuous
at every point within their domain

1. Polynomial Functions
2. Rational Functions
3. Radical Functions
4. Trig Functions

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Continuity of a Composite Function
If g is continuous at c and f is continuous at
g(c), then the composite function given by
is continuous at c.
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Examples 1 2

• Describe the intervals on which each function is
  continuous. Verify graphically.

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Examples 3 4

• Describe the intervals on which each function is
  continuous. Verify graphically.

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Example 5

• Describe the intervals on which each function is
  continuous. Verify graphically.

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Example 6

• Find the constant a such that f(x) is continuous
  on .

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Intermediate Value Theorem (IVT)It aint a
sandwich unless theres something between the
bread.
If f is continuous on the closed interval a, b
and k is any number between f(a) and f(b), then
there is at least one number c in a, b such
that f(c) k.
A continuous function takes on all values between
any two points that it assumes.
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IVT Example

• Consider this example
• A baby weighs 7.3 lbs on day 1 and weighs 8.9
  lbs on day 24. There has to be a time between day
  1 24 when the baby weighed exactly 8.0 lbs.

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Finding Zeros

• So how do we use the IVT?
• Well use it primarily to locate zeros of a
  function that is continuous on an interval.

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Finding Zeros

• Use the Intermediate Value Theorem to show that
  has a zero in the interval 0, 1.
  Then use your calculator to find the zero
  accurate to four decimal places.

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Homework
Section 1.4 page 78 1-17 odd, 25, 29 37 odd,
69, 71, 77, 79
Dont Skip!!!!