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Math

16A

Kouba

Continuity

of

Functions

THREE-STEP

DEFINITION:

Function

y

=

f(z)

is

continuous

at

x

=

a

if

i.)

f(a)

exists

(is

a

finite

number)

ii.)

limf(x)

exists

(is

a

finite

number)

Hi.)

Hmf(x)

=

f(a)

SHORTCUT

1:

Every

polynomial

is

continuous

for

all

values

of

x.

SHORTCUT

2:

Sums,

differences,

products, quotients

(denominator

$

0),

and

compositions

of

continuous

functions

are

continuous.

HERE

is

a

short

list

of

WELL-KNOWN

CONTINUOUS FUNCTIONS:

1.

sinx

(for

all

x-values)

2.

cosx

(for

all

x-values)

3.

/E

(for

all

x

0)

4

‘3

(for

all

x-values)

5.

c

(for

all

x-values)-

formally

introduced

in

Math

16B

6.

lnx

(for

all

x

>

0)

formally

introduced

in

Math

16B

1