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