CalculuS Continuity Lesson 1

Continuity at a Point

Any function whose graph can be sketched in one continuous motion

without lifting the pencil is an example of a continuous function.

y f x

If a function f is not continuous at a point c ,

we say that f is discontinuous at c and c is a

point of discontinuity of f.

Note that c need not be in the domain of f.

CONTINUITY Functions such as f(x) = 2x–1 are considered

continuous which implies that as it graphs there are no skips.

Are these continuous?

1.

2. 3.

4. 5. 6.

2. 2.

6. 5.

Which rule is violated by each of these?

Continuous Functions

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 need not be continuous on

every interval.

Do #4 and 5 p.58 Packet

Continuity is also interfered with in

problems when in function form.

1,4

1,3.3

0,3

0,3.2

1

2.1

x

xxy

xx

xxy

x

xy

Assignment

Finish Packet

Intermediate Value Theorem

Slide 2- 21

Properties of Continuous Functions

If the functions and are continuous at , then the

following combinations are continuous at .

1. Sums :

2. Differences:

3. Products:

4. Constant multiples: , for any number

5. Quotients: , pr

f g x c

x c

f g

f g

f g

k f k

f

g

ovided 0g c

Composite of Continuous Functions

If is continuous at and is continuous at , then the

composite is continuous at .

f c g f c

g f c

Slide 2- 23

Intermediate Value Theorem for Continuous Functions

0 0

A function that is continuous on a closed interval [ , ]

takes on every value between and . In other words,

if is between and , then for some in [ , ].

y f x a b

f a f b

y f a f b y f c c a b

Slide 2- 24

Intermediate Value Theorem for Continuous Functions

The Intermediate Value Theorem for Continuous

Functions is the reason why the graph of a

function continuous on an interval cannot have

any breaks. The graph will be connected, a

single, unbroken curve. It will not have jumps or

separate branches.

Assignment

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IVT 1.4 WS

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