discontinuous

discontinuous

Definition

Suppose

A

is an

open set

in

ℝ

(say an

interval

A

=

(

a

,

b

)

, or

A

=

ℝ

),
and

f

:

A

→

ℝ

is a

function

.
Then

f

is

discontinuous

at

x

∈

A

, if

f

is not

continuous
at

x

. One also says that

f

is discontinuous at all boundary points of

A

.

We know that

f

is continuous at

x

if and only
if

lim

z

→

x

⁡

f

⁢

(

z

)

=

f

⁢

(

x

)

. Thus, from the properties of the

one-sided limits

, which we denote by

f

⁢

(

x

+

)

and

f

⁢

(

x

-

)

, it follows
that

f

is discontinuous at

x

if and only if

f

⁢

(

x

+

)

≠

f

⁢

(

x

)

, or

f

⁢

(

x

-

)

≠

f

⁢

(

x

)

.

If

f

is discontinuous at

x

∈

A

¯

, the

closure

of

A

, we can then distinguish four types of
different

discontinuities

as follows

[

1

,

2

]

:

1.

If

f

⁢

(

x

+

)

=

f

⁢

(

x

-

)

, but

f

⁢

(

x

)

≠

f

⁢

(

x

±

)

,
then

x

is called a

removable discontinuity

of

f

.
If we modify the value of

f

at

x

to

f

⁢

(

x

)

=

f

⁢

(

x

±

)

,
then

f

will become continuous at

x

.
This is clear since the modified

f

(call it

f

0

) satisfies

f

0

⁢

(

x

)

=

f

0

⁢

(

x

+

)

=

f

0

⁢

(

x

-

)

.

2.

If

f

⁢

(

x

+

)

=

f

⁢

(

x

-

)

, but

x

is not in

A

(so

f

⁢

(

x

)

is
not defined), then

x

is also called a

removable discontinuity

.
If we assign

f

⁢

(

x

)

=

f

⁢

(

x

±

)

, then this modification renders

f

continuous at

x

.

3.

If

f

⁢

(

x

-

)

≠

f

⁢

(

x

+

)

, then

f

has a

jump discontinuity

at

x

Then the number

f

⁢

(

x

+

)

-

f

⁢

(

x

-

)

is then called the

jump

,
or

saltus

, of

f

at

x

.

4.

If either (or both) of

f

⁢

(

x

+

)

or

f

⁢

(

x

-

)

does not exist, then

f

has an

essential discontinuity

at

x

(or a

discontinuity of the second kind

).

Note that

f

may be

continuous

(continuous in all points in

A

), but still have discontinuities in

A

¯

Examples

1.

Consider the function

f

:

ℝ

→

ℝ

given by

f

⁢

(

x

)

=

{

1

when

⁢

x

≠

0

,

0

when

⁢

x

=

0

.

Since

f

⁢

(

0

-

)

=

1

,

f

⁢

(

0

)

=

0

, and

f

⁢

(

0

+

)

=

1

,
it follows that

f

has a removable discontinuity at

x

=

0

.
If we modify

f

⁢

(

0

)

so that

f

⁢

(

0

)

=

1

, then

f

becomes the

continuous function

f

⁢

(

x

)

=

1

.

2.

Let us consider the function defined by the formula

f

⁢

(

x

)

=

sin

⁡

x

x

where

x

is a nonzero

real number

. When

x

=

0

, the formula is undefined, so

f

is only determined for

x

≠

0

. Let us show that this point is
a removable discontinuity. Indeed, it is easy
to see that

f

is continuous for all

x

≠

0

, and using
L’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

) we have

f

⁢

(

0

+

)

=

f

⁢

(

0

-

)

=

1

.
Thus, if we assign

f

⁢

(

0

)

=

1

, then

f

becomes a continuous function
defined for all real

x

. In fact,

f

can be made into an

analytic
function

on the whole

complex plane

.

3.

The

signum function

sign

:

ℝ

→

ℝ

is defined as

sign

(

x

)

=

{

-

1

when

⁢

x

<

0

,

0

when

⁢

x

=

0

,

and

1

when

⁢

x

>

0

.

Since

sign

(

0

+

)

=

1

,

sign

(

0

)

=

0

, and since

sign

(

0

-

)

=

-

1

,
it follows that

sign

has a jump discontinuity at

x

=

0

with jump

sign

(

0

+

)

-

sign

(

0

-

)

=

2

.

4.

The function

f

:

ℝ

→

ℝ

given by

f

⁢

(

x

)

=

{

1

when

⁢

x

=

0

,

sin

⁡

(

1

/

x

)

when

⁢

x

≠

0

has an essential discontinuity at

x

=

0

. See

[

2

]

for details.

General Definition

Let

X

,

Y

be

topological spaces

, and let

f

be a mapping

f

:

X

→

Y

. Then

f

is

discontinuous

at

x

∈

X

, if

f

is not
continuous at (

http://planetmath.org/Continuous

)

x

.

In this generality, one generally does not classify discontinuities quite so closely, since they can have quite complicated behaviour.

Notes

A jump discontinuity is also called a

simple discontinuity

, or a

discontinuity of the first kind

.
An

essential discontinuity

is also called a

discontinuity of the second kind

.

References

1

R.F. Hoskins,

Generalised functions

,
Ellis Horwood Series: Mathematics and its applications,
John Wiley & Sons, 1979.

2

P. B. Laval,

http://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdf

http://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdf.

Title

discontinuous

Canonical name

Discontinuous

Date of creation

2013-03-22 13:45:01

Last modified on

2013-03-22 13:45:01

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

14

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A15

Classification

msc 54C05

Defines

removable discontinuity

Defines

saltus

Defines

jump

Defines

jump discontinuity

Defines

discontinuity of the second kind

Defines

discontinuity of the first kind

Defines

essential discontinuity

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