Objectives:1. Be able to define continuity by determining if a graph is

continuous.2. Be able to identify and find the different types of

discontinuities that functions may contain.3. Be able to determine if a function is continuous on a

closed interval.4. Be able to determine one-sided limits and continuity on

a closed interval.

Critical Vocabulary:Limit, Continuous, Continuity, Composite Function

I. Continuity

Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c

This means a graph will contain no _____, _____, or _____

Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.

I. Continuity

What Causes discontinuity?

1. The function is not defined at c.

This is an example of a ____ in the graph at ___

Concept: The function is not defined at c.

__________________

Let’s look at at f(x) = ½x - 2

I. Continuity

What Causes discontinuity?

2. The limit of f(x) does not exist at x = c

This is an example of a ____ in the graph at ______

Concept: The limit does not exist at x = c

Let’s look at at

37

313

1)(

xx

xxxf

I. Continuity

What Causes discontinuity?

3. The limit of f(x) exists at x = c but is not equal to f(c).

This is an example of a _____ in the graph

Concept: The behavior (limit) and where its defined (f(c)) are __________________

Let’s look at the first graph again

)()(lim cfxfcx

What is the limit as x approaches -2?

What is f(-2)?

A function f is continuous at c if the following three conditions are met:

1. ___________________________

2. ___________________________

3. ___________________________

I. Continuity

Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c

This means a graph will contain no HOLES, JUMPS, or GAPS

Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.

Objectives:1. Be able to define continuity by determine if a graph is

continuous.2. Be able to identify and find the different types of

discontinuities that functions may contain.3. Be able to determine if a function is continuous on a

closed interval.4. Be able to determine one-sided limits and continuity on

a closed interval.

Critical Vocabulary:Limit, Continuous, Continuity, Composite Function

II. Discontinuities

When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.

Discontinuity is broken into 2 Categories:

1. Removable:A discontinuity is removable if you COULD define f(c).

II. Discontinuities

When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.

Discontinuity is broken into 2 Categories:

1. Removable:A discontinuity is removable if you COULD define f(c).

2. Non-Removable:

A discontinuity is non-removable if you CANNOT define f(c).

II. Discontinuities

When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.

Discontinuity is broken into 2 Categories:

1. Removable:A discontinuity is removable if you COULD define f(c).

2. Non-Removable:

A discontinuity is non-removable if you CANNOT define f(c).

Example 1:1

1)(

2

x

xxf What is the Domain?

II. Discontinuities

When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.

Discontinuity is broken into 2 Categories:

A discontinuity is removable if you COULD define f(c).

A discontinuity is non-removable if you CANNOT define f(c).

Example 2: 9

3)(

2

x

xxf What is the Domain?

II. Discontinuities

When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.

Discontinuity is broken into 2 Categories:

1. Removable:A discontinuity is removable if you COULD define f(c).

2. Non-Removable:

A discontinuity is non-removable if you CANNOT define f(c).

Example 3: Discuss the continuity of the composite function f(g(x))

xxf

1)( 1)( xxg

II. Discontinuities

Example 5: Graph the piecewise function, then determine on which intervals the graph is continuous.

02

01)( 2 xx

xxxf

III. Closed Intervals

Example 5: Discuss the continuity on the closed interval.

2,14

1)(

2

x

xf

Closed Interval: Focusing on specific portion (domian) of a graph. [a, b]

1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77

2. Page 236 #1-17 odd, 79, 88

Objectives:1. Be able to define continuity by determine if a graph is

continuous.2. Be able to identify and find the different types of

discontinuities that functions may contain.3. Be able to determine if a function is continuous on a

closed interval.4. Be able to determine one-sided limits and continuity on

a closed interval.

Critical Vocabulary:Limit, Continuous, Continuity, Composite Function

IV. One-Sided Limits

What does a One-Sided look like?

Lxfcx

)(lim c

Lxfcx

)(lim c Approach from the right only

Lxfcx

)(lim c Approach from the left only

IV. One-Sided Limits

Example 1: Graph then find the limits

24)( xxf

What’s the domain?

x

f(x)

____)(lim2

xfx

____)(lim2

xfx

____)(lim0

xfx

____)(lim2

xfx

____)(lim2

xfx

____)(lim2

xfx

____)(lim2

xfx

IV. One-Sided Limits

Example 1: Graph then find the limits

14

14)( 2 xxx

xxxf

x

f(x)

____)(lim1

xfx

____)(lim1

xfx

____)(lim1

xfx

Is this graph continuous?

1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77

2. Page 236 #1-17 odd, 79, 88