5_2.4-Continuity.pdf

AP Calculus

Sec 2.4 Continuity

In many cases, you can compute lim ( )

x!a f x by plugging a in for x: lim ( ) ( )

x!a f x " f a

For example, 3 3 .

3

lim(2 5 1) 2 3 5 3 1 40

x! x # x$ " % # % $ "

This situation arises often enough that it has a name.

1.

Definition

A function ( )f x is continuous at a if lim ( ) ( ) x!a f x " f a .

This definition really comprises three things, each of which you need to check to show that f is continuous at a.

1. f a( ) is ________________.

2. lim ( )

x!a f x is ________________.

3. The two are equal: ____________________________________.

What does this mean geometrically? Here are the three criteria above in pictorial language:

1. “ ( )f a is defined” means _____________________________________________________.

2. “ lim ( )

x!a f x is defined” means __________________________________________________. __________________________________________________________________________.

3. “ lim ( ) ( )

x!a f x " f a ” means _____________________________________________________

__________________________________________________________________________.

The first criterion means that there can’t be a _________ or _________ in the graph. This also rules out vertical asymptotes. Here are some pictures of these kinds of discontinuities:

The second criterion means that the graph can’t __________ at a. This is a jump discontinuity.

The third criterion means that the graph is ___________ at x"a as you’d expect. You don’t get close to a expecting one value and then find that ( )f a is something different as you do below.

2.

Continuity at Endpoints

A function y" f x

& '

continuous at left endpoint a

or is

continuous at right endpoint b

Example 1 – Investigating Continuity

1 2 3 4 1

2

0

y" f x

& '

Find the points at which the function f is continuous and the points at which f is discontinuous.

What is the relationship between the limit of f and the value of f at each point where the graph is either continuous or discontinuous?

3.

Continuous Functions

The fine print: A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function does not need to be continuous on every interval. For example,

y" 1

x is not continuous on

(

)

State some functions that you know that are continuous everywhere (#* *, ).

4.

Algebraic Combinations

You can use operations on functions to create new continuous functions.

Properties of Continuous Functions

+ If f and g are continuous at x "c, so is their sum + If f and g are continuous at x "c, so is their difference + If f and g are continuous at x "c, so is their product

+ If f and g are continuous at x "c, and if g c

& '

+ If f is continuous at g c( ) and if g is continuous at x"c, then the composite

Example 2

Since sinx and x3 are continuous for all x, then

Composition is an important method for constructing continuous functions. For example, ( ) sin

f x " x is continuous for all x. The polynomial g x( )"x4#7x2$ $x 1 is also continuous for all x. Hence the composite

Example 3 – Given

2

1, 0 ( ) , 0 1

( 1) 1, 1 x

f x x x

x x

- .

/ " 0

/ # $ 1

2

3 . , where is ( )f x continuous?

First graph the function. Determine the points of continuity…

Example 4 -If a function is continuous at a, we can substitute to find the limit:

a)

&

'

2

lim 5 7 b)

x! x # x $x #

&

'

2 0

lim 3 2

x! x # x$

Example 5 – Given ( ) 2 ₂2

1

x x

f x

x

# #

"

# 3

, where is ( )f x continuous?

In the above example, the graph of f approaches 2 near x" #1 but at x" #1 there is a hole. You can plug the hole if you artificially define ( 1)f # "2. Then, f would be continuous at x" #1. In this situation, we say that x" #1 is a ________________________.

The discontinuity at x"1 is a vertical asymptote no matter how we define f(1) so the function will still be discontinuous at x"1. x"1 represents a _________________________________.

Example 6 – Find the points of discontinuity and state if removable.

a) ( ) 3 5 f x

x "

$ b) ( ) 9

x

f x " c)

2 1 ( ) 1 x f x x # " #

d) ( ) ₂5 3 f x

x "

$ e)

3 , 2 ( )

1, 2 2

x x

f x _x

x # 3 -/ " 0 $ 4 /2

Example 7 – Find a value for k so that the function is continuous.

2

1, 3 ( )

2 , 3

x x f x kx x - # . " 0 1 2

5.

Intermediate Value Theorem for Continuous Functions

Continuous functions possess the intermediate value property. Roughly put, it says that if a continuous function goes from one value to another, it doesn’t skip any values in between. This corresponds to the geometric intuition that the graph of a continuous function doesn’t have any gaps, jumps or holes. Here is the precise statement:

y

Calculus Section 2.4 Page 6 of 6

x

One subtle point which you should understand is the following:

You can know something exists without being able to find it.

If I take your house keys and throw them into a nearby corn field, you know that your keys are in the field – but finding them is a different story!

The Intermediate Value Theorem says there is a number c such that ( )f c "m. It doesn’t tell you how to find it, though you can usually approximate c as closely as you want.

And by the way, there may be more than one number c which works.

Example 8 – Using the Intermediate Value Theorem