Continuity

Lesson 1.1.14

Learning Objectives

• Given a function, determine if it is continuous at a certain point using the three criteria for continuity.

• Evaluate a one-sided limit.

• Determine if a function is continuous on a closed interval using one-sided limits.

• Apply the above objectives to the greatest integer function.

Continuity: The Big Picture

• Continuous functions are the “normal” functions. They are the ones that do not have any holes, asymptotes, jumps, or matter that break the smooth flow of the function.

• The functions on the right are continuous throughout

y = x

y = x2

More Continuous Functions

y = sin x y = x3

y = 3y = ex

Which are continuous throughout?

Continuity at a Point

• The functions on the right are not continuous at x = 0, but they are continuous at all of the other points.

• You can usually tell by looking if a function is continuous at a certain point.

• However, in calculus, we have a more formal definition of continuity.

Definition of Continuous

• A function f(x) is continuous at x = c if:

• The functions on the right are y = 1/x and y = |x|/x.

• For both functions, f(0) does not exist, for it is undefined.

• Therefore, the functions are discontinuous at x = 0

Pre-Example 1

• Observe the function on the right. Let’s see if it is continuous at x = 1.

• It certainly meets the first criteria, for f(1) exists. In fact, f(1) = 1.

• However, the limit as x1 does not exist.

• Therefore, the function is not continuous at x = 1

Pre-Example 2

• Observe the function on the right. Is it continuous at x = 1?

• First criteria: f(1) = 3. f(1) exists!

• The limit as x1 is 1. The limit exists too!

• However, f(1) and the limit do not equal. Fails third criteria.

Example 1

Determine if the following function is continuous at h(1) and h(2).

• (Use your three criteria.)

One-sided Limits

We’ve learned:

The limit as x approaches c from the left.

The limit as x approaches c from the right.

)(lim xfcx

The overall limit. This exists only if the left and right limits are equal.

• In this lesson, you will get problems that ask you to evaluate one-sided limits.

Example 2

• Evaluate:

x

xx 0lim

x

xx 0lim

The Greatest Integer Function

• This is also known as the step function. You can probably see why in the graph of y = ||x|| on the right.

• To evaluate the above function for a given x, determine the integer that is equal to or just below x.

For example:• ||3.7|| = 3• ||4|| = 4• ||-5.9|| = 6

• The greatest integer function is one in which one-sided limits are especially used.

• For example, what is:

xx 2lim

xx 2lim

Interval Review

Remember:

• (a, b) is called an open interval. It is the range of every single number from a to b, not including a and b.

• [a, b] is called a closed interval. It is like an open interval except that it includes a and b.

Continuity on an Open Interval

• A function is continuous on the open interval (a, b) if it is continuous at every point in that interval.

Continuity on a Closed Interval

A function is continuous on the closed interval [a, b] if:

• It is continuous on the open interval (a, b)

• The following one-sided limits exist:

Example 3

Determine if the function below is continuous on the interval [-1, 1].

Remember:

• First determine if it is continuous on (-1, 1) (You can do this by graphing.)

• Then evaluate the one-sided limits at -1 and 1.

Wrap-Up

• Know how to determine if a function is continuous at a point using the three criteria.

• Know one-sided limits.

• Know the greatest integer function.

• Know how to determine continuity on intervals.

Homework

• Reteaching