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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

c

Continuous Function

Continuity at a Pointfis 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.

1)f is not continuous at c because

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 c.

3)f is not continuous at c because

f(c)

a

c

b

Continuity on an Open IntervalA 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

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

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

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

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

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 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.*

Example 3

is called the ______________ __________________ ______________.

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 IntervalA function is continuous on a closed interval if it is continuous everywhere inside the interval and has one-sided continuity at the endpoints.

Examples

Discuss the continuity of the function on the closed interval.

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

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 & 2

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

Examples 3 & 4

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

Example 5

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

Example 6

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

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.

Weight (lbs) tells you one exists. This is an

Day

IVT ExampleConsider 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 tells you one exists. This is an

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 Zeros tells you one exists. This is an

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 tells you one exists. This is an

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

Don’t Skip!!!!

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