continuity grand canyon, arizona greg kelly, hanford high school, richland, washingtonphoto by...

Continuity

Grand Canyon, ArizonaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

A function is continuous at a point if the limit is the same as the value of the function.

This function has discontinuities at x=1 and x=2.

It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

1 2 3 4

1

2

jump infinite oscillating

Essential Discontinuities:

Removable Discontinuities:

(You can fill the hole.)

Removing a discontinuity:

3

2

1

1

xf x

x

has a discontinuity at .1x

Write an extended function that is continuous at .1x

3

21

1lim

1x

x

x

2

1

1 1lim 1 1x

x x xx x

1 1 1

2

3

2

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.

1x

Removing a discontinuity:

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.

1x

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

Also: Composites of continuous functions are continuous.

examples: 2siny x cosy x

Definition of Continuity

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

1. f c is defined

2. limx cf x exists

3. limx cf x f c

A function is continuous on an open interval

If it is continuous at each point in the interval. A

function that is continuous on the entire real line is everywhere

continuous

,a b

Formal Definition of Continuityon an Interval

A function f is continuous on a closed interval

,a b

,a b

If it is continuous on the open interval and

limx a

f x f a

and limx b

f x f b

The function f is continuous from the right at

and continuous from the left at

ab

Intermediate Value Theorem (IVT)

If a function is continuous between a and b, then it takes

on every value between and . f a f b

a b

f a

f b

Because the function is continuous, it must take on every y value between and .

f a f b

Example 5: Is any real number exactly one less than its cube?

(Note that this doesn’t ask what the number is, only if it exists.)

3 1x x

30 1x x

3 1f x x x

1 1f 2 5f

Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5.Therefore there must be at least one solution between 1 and 2.

Use your calculator to find an approximate solution.

3

1

2 1

Y x

Y x

or 31 1Y x x

Graphing calculators can make non-continuous functions appear continuous.

The calculator “connects the dots” which covers up the discontinuities.

Use your calculator to graph the function

1

2

xf x

x

Graphing calculators can make non-continuous functions appear continuous.

Graph: intf x x

GREATEST INTEGER FUNCTION

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Graphing calculators can make non-continuous functions appear continuous.

If we change the plot style to “dot”, we get a graph that is closer to the correct graph of the function.

The open and closed circles do not show, but we can see the discontinuities.

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

-1

-2

-3

-4

-5

x

y