Continuity one sided limits
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Continuity & One-Sided Limits. Section 1.4. 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. f. c.

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Continuity & One-Sided Limits

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Continuity one sided limits

Continuity & One-Sided Limits

Section 1.4


After this lesson you will be able to

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


Continuity at a point

f

c

Continuous Function

Continuity at a Point

fis 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.


Some examples of functions that are not continuous at c

Some examples of functions that are NOT continuous at c.

1)f is not continuous at c because


Some examples of functions that are not continuous at c1

Some examples of functions that are NOT continuous at c.

2)f is not continuous at c because


Some examples of functions that are not continuous at c2

Some examples of functions that are NOT continuous at c.

3)f is not continuous at c because


Continuity on an open interval

f

f(c)

a

c

b

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.


Discontinuity

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.


Removable discontinuity

Removable Discontinuity

f

f(c)

Removable discontinuity at x=c, since f can be made continuous by just redefining f(c).

c

Discontinuity at x = c


Non removable discontinuity

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


Continuity

Continuity

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


One sided limits

One-sided Limits

Limit from the right(right-hand limit)

Limit from the left(left-hand limit)


One sided limits1

One-sided Limits

*One-sided limits are great for radical functions.*

Example:

We can’t 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.


One sided limits2

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.*


Examples 1 2

Examples 1 & 2


Example 3

Example 3

is called the ______________ __________________ ______________.


Continuity on a closed interval

Example: Find the one-sided limits at the endpoints of the function,

At –2, we can only take a right limit:

2

-2

At 2, we can only take a left limit:

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.


Examples

Examples

Discuss the continuity of the function on the closed interval.


Properties of continuity

Properties of Continuity

If k is a real number and f and g are continuous at x = c, then f + g, f-g, f•g, kf, and f/g(provided g(c)  0) are also continuous.

The following types of functions are continuous at every point within their domain:

  • Polynomial Functions

  • Rational Functions

  • Radical Functions

  • Trig Functions


Continuity of a composite function

We can also say,

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.


Examples 1 21

Examples 1 & 2

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


Examples 3 4

Examples 3 & 4

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


Example 5

Example 5

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


Example 6

Example 6

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


Intermediate value theorem ivt it ain t a sandwich unless there s something between the bread

This theorem doesn’t tell you the value of c, but just tells you one exists. This is an existence theorem.

f(b)

k

f(a)

a

c

b

f(c) = k

Intermediate Value Theorem (IVT)“It ain’t a sandwich unless there’s 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.


Ivt example

Weight (lbs)

Day

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.


Finding zeros

Finding Zeros

So how do we use the IVT?

We’ll use it primarily to locate zeros of a function that is continuous on an interval.


Finding zeros1

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.


Homework

Homework

Section 1.4:page 78 #1-17 odd, 25, 29 – 37 odd, 69, 71, 77, 79

Don’t Skip!!!!


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