aarjaneiro/planetmath huggingface dataset:
envelope of a function

envelope of a function

Consider

f

:

ℝ

→

ℝ

a

real function

of real variable.

We call the upper envelope of

f

to the

function

:

env

sup

⁡

(

f

)

⁢

(

x

)

=

inf

ϵ

⁡

{

sup

⁡

{

f

⁢

(

y

)

:

ϵ

>

0

,

|

y

-

x

|

<

ϵ

}

}

similarly the lower envelope of

f

is the function:

env

inf

⁡

(

f

)

⁢

(

x

)

=

sup

ϵ

⁡

{

inf

⁡

{

f

⁢

(

y

)

:

ϵ

>

0

,

|

y

-

x

|

<

ϵ

}

}

The envelopes have the following properties:
(in this list

env

∗

represents either the upper or lower envelope)

•

env

inf

⁡

(

f

)

⁢

(

x

)

≤

f

⁢

(

x

)

≤

env

sup

⁡

(

f

)

⁢

(

x

)

•

env

sup

⁡

(

f

)

=

env

inf

⁡

(

f

)

⇔

f

⁢

is continuous

•

env

sup

⁡

(

f

)

⁢

(

x

)

-

env

inf

⁡

(

f

)

⁢

(

x

)

=

oscillation of

⁢

f

⁢

at

⁢

x

•

env

inf

⁡

f

=

-

env

sup

⁡

(

-

f

)

Title

envelope of a function

Canonical name

EnvelopeOfAFunction

Date of creation

2013-03-22 15:44:22

Last modified on

2013-03-22 15:44:22

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

5

Author

cvalente (11260)

Entry type

Definition

Classification

msc 26A99

Generated on Fri Feb 9 19:55:26 2018 by

LaTeXML
piecewise

piecewise

The word “

piecewise

” is used widely in mathematics, primarily in the

analysis

of

functions

of a single real variable. Piecewise is typically applied to a set of mathematical properties on a function. Loosely speaking, a function

satisfies

a particular property “piecewise” if that function can be broken down into pieces (to be made precise later) so that each piece satisfies that particular property. However, to avoid

potential

problems with

infinity

, the number of pieces is generally set to be finite (particularly in the case when the domain is bounded). Another potential problem is that the function having this “piecewise” property (let’s call it

P

) usually fails to have this property

P

at certain boundary points of the pieces. To get around this technicality, and thus allowing a much wider class of functions to being “piecewise

P

”, pieces are re-defined so as to exclude these problematic “boundary points”.

Formally speaking, we have the following:

That a function

f

with domain

D

⊆

ℝ

having “piecewise” property

P

means that there is a finite

partition

of

D

:

D

=

D

1

∪

⋯

∪

D

n

,

with

⁢

D

i

∩

D

j

=

∅

⁢

for

⁢

i

≠

j

such that the

restriction

of

f

to the interior of each

D

i

:

f

i

:=

f

∣

int

(

D

i

)

has property

P

.

Remarks

.

•

If

D

is an

interval

or a ray on

ℝ

, then this finite partition can usually be done so that each “piece” is an interval or a ray.

•

If function

f

satisfies property

P

, then

f

satisfies

P

piecewise.

•

Conversely

, if

f

satisfies property

P

piecewise and

f

satisfies

P

at the boundary points of each “piece” of the domain

D

, then

f

satisfies

P

.

For example, if

P

means

continuity

of a function, then to say that a function

f

defined on

ℝ

is

piecewise continuous

is the same thing as saying that

ℝ

can be partitioned into intervals and rays so that

f

is

continuous

in each of the intervals and rays.

Other commonly used terms involving the

concept

of “piecewise” are

piecewise

differentiable

,

piecewise smooth

,

piecewise linear

, and

piecewise

constant

.

Anyone who can supply some graphs illustrating the concepts mentioned above will be greatly appreciated.

Title

piecewise

Canonical name

Piecewise

Date of creation

2013-03-22 15:50:42

Last modified on

2013-03-22 15:50:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 26A99

Generated on Fri Feb 9 19:55:37 2018 by

LaTeXML
power function

power function

A real

power function

f

:

ℝ

+

→

ℝ

has the form

f

⁢

(

x

)

=

x

a

where

a

is a given

real number

.

Theorem

.

The power function

x

↦

x

a

is

differentiable

with the

derivative

x

↦

a

⁢

x

a

-

1

and

strictly increasing

if

a

>

0

and

strictly decreasing

if

a

<

0

(and 1 if

a

=

0

).

The power functions comprise the

natural power functions

x

↦

x

n

with

n

=

0

,

1

,

2

,

…

,  the

root functions

x

↦

x

n

=

x

1

n

with

n

=

1

,

2

,

3

,

…

and other

fraction power functions

x

↦

x

a

with

a

any fractional number.

Note.

The power

x

a

may of course be meaningful also for other than positive values of

x

, if

a

is an integer.  On the other hand, e.g.

(

-

1

)

2

has no real values — see the general power.

Title

power function

Canonical name

PowerFunction

Date of creation

2013-03-22 14:46:32

Last modified on

2013-03-22 14:46:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A99

Related topic

PropertiesOfTheExponential

Related topic

FractionPower

Related topic

CubeOfANumber

Related topic

Polytrope

Related topic

PowerTowerSequence

Related topic

LaplaceTransformOfLogarithm

Defines

natural power function

Defines

root function

Defines

fraction power function

Generated on Fri Feb 9 19:55:38 2018 by

LaTeXML
real function

real function

A

real function

is a

function

from a subset of

ℝ

to

ℝ

.
Without imposing extra conditions, there is really not much one can say about real
functions. Therefore, the usual procedure is to select some subclass of real
functions according to some criterion such as differentiability or integrability
and focus attention on this subclass. There follows a list of such criteria:

•

elementary function

•

increasing function

,

decreasing function

•

positive function, negative function

•

Title

real function

Canonical name

RealFunction

Date of creation

2013-12-26 13:42:26

Last modified on

2013-12-26 13:42:26

Owner

rspuzio (6075)

Last modified by

pahio (2872)

Numerical id

9

Author

rspuzio (2872)

Entry type

Topic

Classification

msc 26A99

Related topic

RationalFunction

Related topic

ComplexFunction

Generated on Fri Feb 9 19:55:40 2018 by

LaTeXML
ruler function

ruler function

The

ruler function

f

on the real line is defined as follows:

f

⁢

(

x

)

=

{

0

,

x

is irrational;

1

/

n

,

x

=

m

/

n

,

m

and

n

are relatively primes

.

(1)

Given a

rational number

m

n

in lowest terms,

n

positive

, the ruler function outputs the size (length) of a piece resulting from equally subdividing the unit interval into

n

, the number in the denominator, parts. It “ignores” inputs of irrational

functions

, sending them to 0.

The ruler function is so termed because it resembles a ruler. The following picture might be helpful: if

m

n

in lowest terms is a reasonably small rational number (which we assume positive). Then it can be “read off” on a ruler whose

intervals

of one unit size are each equally subdivided into

n

parts measuring

1

n

units each by

1.

running one’s finger through until the integer preceding it and then

2.

running through to the subsequent

r

th subunit, “left-over” from the division of

m

by

n

.

On the other hand, an irrational number can not be read off from any ruler no matter how fine we subdivide a unit interval in any ruler.

References

1

Dunham, W.,

Nondifferentiability of the Ruler Function

, Mathematics Magazine,

Mathematical Association of America

, 2003.

2

Heuer, G.A.,

Functions

Continuous at

the Irrationals and

Discontinuous

at the Rationals

, The American Mathematical Monthly, Mathematical Association of America, 1965.

Title

ruler function

Canonical name

RulerFunction

Date of creation

2013-03-22 18:23:55

Last modified on

2013-03-22 18:23:55

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

5

Author

yesitis (13730)

Entry type

Definition

Classification

msc 26A99

Generated on Fri Feb 9 19:55:42 2018 by

LaTeXML
alternative proof of derivative of x^n

alternative proof of derivative of

x

n

The typical

derivative

formula

d

⁢

f

d

⁢

x

=

lim

h

→

0

⁡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

combined with the

binomial theorem

yield an alternative way to prove that

d

d

⁢

x

⁢

(

x

n

)

=

n

⁢

x

n

-

1

for any

positive integer

n

.

Proof.

d

d

⁢

x

⁢

(

x

n

)

=

lim

h

→

0

⁡

(

x

+

h

)

n

-

x

n

h

=

lim

h

→

0

⁡

(

∑

j

=

0

n

(

n

j

)

⁢

x

j

⁢

h

n

-

j

)

-

x

n

h

=

lim

h

→

0

⁡

x

n

+

n

⁢

x

n

-

1

⁢

h

+

h

2

⁢

(

∑

j

=

0

n

-

2

(

n

j

)

⁢

x

j

⁢

h

n

-

2

-

j

)

-

x

n

h

=

lim

h

→

0

⁡

n

⁢

x

n

-

1

⁢

h

+

h

2

⁢

∑

j

=

0

n

-

2

(

n

j

)

⁢

x

j

⁢

h

n

-

2

-

j

h

=

lim

h

→

0

⁡

(

n

⁢

x

n

-

1

+

h

⁢

∑

j

=

0

n

-

2

(

n

j

)

⁢

x

j

⁢

h

n

-

2

-

j

)

=

n

⁢

x

n

-

1

∎

Title

alternative proof of derivative of

x

n

Canonical name

AlternativeProofOfDerivativeOfXn

Date of creation

2013-03-22 15:59:31

Last modified on

2013-03-22 15:59:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 26B05

Classification

msc 26A24

Related topic

DerivativeOfXn

Related topic

DerivativesByPureAlgebra

Generated on Fri Feb 9 19:55:44 2018 by

LaTeXML
C_0^∞⁢(U) is not empty

C

0

∞

⁢

(

U

)

is not empty

Theorem.

If

U

is a non-empty

open set

in

ℝ

n

, then the set of

smooth functions with compact support

C

0

∞

⁢

(

U

)

is non-trivial (that is, it contains

functions

other than the

zero function

).

Remark.

This theorem may seem to be obvious at first sight.
A way to notice that it is not so obvious, is to formulate it for

analytic functions

with compact support: in that case, the result
does not hold; in fact, there are no nonconstant analytic
functions with compact support at all.
One important consequence of this theorem is the

existence of partitions
of unity

.

Proof of the theorem:

Let us first prove this for

n

=

1

:
If

a

<

b

be

real numbers

, then there exists a
smooth non-negative function

f

:

ℝ

→

ℝ

, whose

support

(

http://planetmath.org/SupportOfFunction

) is the

compact set

[

a

,

b

]

.

To see this, let

ϕ

:

ℝ

→

ℝ

be the function
defined on this page (

http://planetmath.org/InfinitelyDifferentiableFunctionThatIsNotAnalytic

),
and let

f

⁢

(

x

)

=

ϕ

⁢

(

x

-

a

)

⁢

ϕ

⁢

(

b

-

x

)

.

Since

ϕ

is smooth, it follows that

f

is smooth. Also, from the
definition of

ϕ

, we see that

ϕ

⁢

(

x

-

a

)

=

0

precisely when

x

≤

a

, and

ϕ

⁢

(

b

-

x

)

=

0

precisely when

x

≥

b

.
Thus the support of

f

is indeed

[

a

,

b

]

.

Since

U

is non-empty and
open there exists an

x

∈

U

and

ε

>

0

such that

B

ε

⁢

(

x

)

⊆

U

. Let

f

be

smooth function

such that

supp

⁡

f

=

[

-

ε

/

2

,

ε

/

2

]

, and
let

h

⁢

(

z

)

=

f

⁢

(

∥

x

-

z

∥

2

)

.

Since

∥

⋅

∥

2

(Euclidean norm) is smooth, the claim follows.

□

Title

C

0

∞

⁢

(

U

)

is not empty

Canonical name

Cinfty0UIsNotEmpty

Date of creation

2013-03-22 13:43:57

Last modified on

2013-03-22 13:43:57

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

17

Author

matte (1858)

Entry type

Theorem

Classification

msc 26B05

Generated on Fri Feb 9 19:55:46 2018 by

LaTeXML
derivative for parametric form

derivative for parametric form

Instead of the usual way

y

=

f

⁢

(

x

)

to present plane curves it is in many cases more comfortable to express both coordinates,

x

and

y

, by means of a suitable

auxiliary variable

, the parametre. It is true e.g. for the cycloid curve.

Suppose we have the parametric form

x

=

x

⁢

(

t

)

,

y

=

y

⁢

(

t

)

.

(1)

For getting now the

derivative

d

⁢

y

d

⁢

x

in a point

P

0

of the curve, we chose another point

P

of the curve. If the values of the parametre

t

corresponding these points are

t

0

and

t

, we thus have the points

(

x

⁢

(

t

0

)

,

y

⁢

(

t

0

)

)

and

(

x

⁢

(

t

)

,

y

⁢

(

t

)

)

and the slope of the secant line through the points is the difference quotient

y

⁢

(

t

)

-

y

⁢

(

t

0

)

x

⁢

(

t

)

-

x

⁢

(

t

0

)

=

y

⁢

(

t

)

-

y

⁢

(

t

0

)

t

-

t

0

x

⁢

(

t

)

-

x

⁢

(

t

0

)

t

-

t

0

.

(2)

Let us assume that the

functions

(1) are

differentiable

when

t

=

t

0

and that

x

′

⁢

(

t

0

)

≠

0

. As we let

t

→

t

0

, the left side of (2) tends to the derivative

d

⁢

y

d

⁢

x

and the side to the

quotient

y

′

⁢

(

t

0

)

x

′

⁢

(

t

0

)

. Accordingly we have the result

(

d

⁢

y

d

⁢

x

)

t

=

t

0

=

y

′

⁢

(

t

0

)

x

′

⁢

(

t

0

)

.

(3)

Note that the (3)
may be written

d

⁢

y

d

⁢

x

=

d

⁢

y

d

⁢

t

d

⁢

x

d

⁢

t

.

Example.

For the cycloid

x

=

a

⁢

(

φ

-

sin

⁡

φ

)

,

y

=

a

⁢

(

1

-

cos

⁡

φ

)

,

we obtain

d

⁢

y

d

⁢

x

=

d

d

⁢

φ

⁢

(

1

-

cos

⁡

φ

)

d

d

⁢

φ

⁢

(

φ

-

sin

⁡

φ

)

=

sin

⁡

φ

1

-

cos

⁡

φ

=

cot

⁡

φ

2

.

Title

derivative for parametric form

Canonical

name

DerivativeForParametricForm

Date of creation

2013-03-22 17:30:48

Last modified on

2013-03-22 17:30:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Derivation

Classification

msc 26B05

Classification

msc 46G05

Classification

msc 26A24

Related topic

GoniometricFormulae

Related topic

CurvatureOfNielsensSpiral

Related topic

Parameter

Generated on Fri Feb 9 19:55:55 2018 by

LaTeXML
higher order derivatives

higher order derivatives

Let the

real function

f

be defined and

differentiable

on the

open interval

I

.  Then for every

x

∈

I

,  there exists the value

f

′

⁢

(

x

)

as a certain

real number

.  This means that we have a new

function

x

↦

f

′

⁢

(

x

)

,

(1)

the so-called

derivative function

of

f

; it is denoted by

f

′

:

I

→

ℝ

or simply

f

′

.

Forming the derivative function of a function is called

differentiation

, the corresponding verb is

differentiate

.

If the derivative function

f

′

is differentiable on

I

, then we have again a new function, the derivative function of the derivative function of

f

, which is denoted by

f

′′

.  Then

f

is said to be

twice differentiable

.  Formally,

f

′′

⁢

(

x

)

=

lim

h

→

0

⁡

f

′

⁢

(

x

+

h

)

-

f

′

⁢

(

x

)

h

for

⁢

all

⁢

x

∈

I

.

The function

x

↦

f

′′

⁢

(

x

)

is called the or

the second derivative

of

f

.  Similarly, one can call (1) the of

f

.

Example.

The

first derivative

of

x

↦

x

3

is

x

↦

3

⁢

x

2

and the second derivative is

x

↦

6

⁢

x

,  since

d

d

⁢

x

⁢

(

3

⁢

x

2

)

=

2

⋅

3

⁢

x

2

-

1

=

6

⁢

x

.

If also

f

′′

is a differentiable function, its derivative function is denoted by

f

′′′

and called the of

f

, and so on.

Generally,

f

can have the

derivatives

of first, second, third, …,

n

th order, where

n

may be an arbitrarily big

positive integer

.  If

n

is four or

greater

, the

n

th derivative of

f

is usually denoted by

f

(

n

)

.  In , it’s sometimes convenient to think that the

0

th order derivative

f

(

0

)

of

f

is the function

f

itself.

The phrase “

f

is infinitely differentiable” means that

f

has the derivatives of all .

Title

higher order derivatives

Canonical name

HigherOrderDerivatives

Date of creation

2013-03-22 16:46:30

Last modified on

2013-03-22 16:46:30

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 26B05

Classification

msc 26A24

Related topic

HigherOrderDerivativesOfSineAndCosine

Related topic

TaylorSeriesOfHyperbolicFunctions

Defines

derivative function

Defines

first derivative

Defines

second derivative

Defines

order of derivative

Defines

differentiation

Defines

differentiate

Defines

twice differentiable

Generated on Fri Feb 9 19:56:05 2018 by

LaTeXML
higher order derivatives of sine and cosine

higher order derivatives of sine and cosine

One may consider the sine and cosine either as real (

http://planetmath.org/RealFunction

) or

complex functions

.  In both cases they are everywhere smooth, having the

derivatives

of all orders (

http://planetmath.org/OrderOfDerivative

) in every point.  The formulae

d

n

d

⁢

x

n

⁢

sin

⁡

x

=

sin

⁡

(

x

+

n

⋅

π

2

)

and

d

n

d

⁢

x

n

⁢

cos

⁡

x

=

cos

⁡

(

x

+

n

⋅

π

2

)

,

where

n

=

0

,

1

,

2

,

…

(the derivative of the

0

th

order means the

function

itself), can be proven by induction on

n

.  Another possibility is to utilize Euler’s formula, obtaining

d

n

d

⁢

x

n

⁢

cos

⁡

x

+

i

⁢

d

n

d

⁢

x

n

⁢

sin

⁡

x

=

d

n

d

⁢

x

n

⁢

e

i

⁢

x

=

e

i

⁢

x

⁢

i

n

=

e

i

⁢

x

+

i

⁢

n

⁢

π

2

=

cos

⁡

(

x

+

n

⋅

π

2

)

+

i

⁢

sin

⁡

(

x

+

n

⋅

π

2

)

;

here one has to compare the real (

http://planetmath.org/ComplexFunction

) and

imaginary parts

– supposing that

x

is real.

Title

higher order derivatives of sine and cosine

Canonical name

HigherOrderDerivativesOfSineAndCosine

Date of creation

2013-03-22 14:45:16

Last modified on

2013-03-22 14:45:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Derivation

Classification

msc 26B05

Classification

msc 46G05

Classification

msc 26A24

Related topic

FractionalDifferentiation

Related topic

HigherOrderDerivatives

Related topic

ExampleOfTaylorPolynomialsForSinX

Related topic

CosineAtMultiplesOfStraightAngle

Generated on Fri Feb 9 19:56:07 2018 by

LaTeXML
logarithmic derivative

logarithmic derivative

Given a

function

f

, the quantity

f

′

/

f

is known as the

logarithmic derivative

of

f

. This name comes
from the observation that, on account of the

chain
rule

,

(

log

⁡

f

)

′

=

f

′

⁢

log

′

⁡

(

f

)

=

f

′

/

f

.

The logarithmic derivative has several basic properties
which make it useful in various

contexts

.

The logarithmic derivative of the

product of
functions

is the sum of their logarithmic
derivatives. This follows from the

product rule

:

(

f

⁢

g

)

′

f

⁢

g

=

f

⁢

g

′

+

f

′

⁢

g

f

⁢

g

=

f

′

f

+

g

′

g

The logarithmic derivative of the quotient of
functions is the

difference

of their logarithmic
derivatives. This follows from the

quotient rule

:

(

f

/

g

)

′

f

/

g

=

f

′

⁢

g

-

f

⁢

g

′

g

2

⁢

g

f

=

f

′

f

-

g

′

g

The logarithmic derivative of the

p

-th power
of a function is

p

times the logarithmic
derivative of the function. This follows
from the

power rule

:

(

f

p

)

′

f

p

=

p

⁢

f

p

-

1

⁢

f

′

f

p

=

p

⁢

f

′

f

The logarithmic derivative of the

exponential

of a function equals the

derivative

of a
function. This follows from the chain rule:

(

e

f

)

′

e

f

=

e

f

⁢

f

′

e

f

=

f

′

Using these

identities

, it is rather easy to
compute the logarithmic derivatives of expressions
which are presented in factored form. For

instance

,
suppose we want to compute the logarithmic derivative
of

e

x

2

⁢

(

x

-

2

)

3

⁢

(

x

-

3

)

x

-

1

.

Using our identities, we find that its logarithic
derivative is

2

⁢

x

+

3

x

-

2

+

1

x

-

3

-

1

x

-

1

.

Title

logarithmic derivative

Canonical name

LogarithmicDerivative

Date of creation

2013-03-22 16:47:02

Last modified on

2013-03-22 16:47:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 26B05

Classification

msc 46G05

Classification

msc 26A24

Related topic

ZeroesOfDerivativeOfComplexPolynomial

Generated on Fri Feb 9 19:56:09 2018 by

LaTeXML
normal line

normal line

A

normal line

(or simply

normal

or

perpendicular

) of a curve at one of its points

P

is the line

passing through

this point and perpendicular to the

tangent line

of the curve at

P

.  The point

P

is the

foot

of the normal.

If the

plane curve

y

=

f

⁢

(

x

)

has a skew

tangent

at the point

(

x

0

,

f

⁢

(

x

0

)

)

,  then the slope of the tangent at that point is

f

′

⁢

(

x

0

)

and the slope of the normal at that point is

-

1

f

′

⁢

(

x

0

)

.  The equation of the normal is thus

y

-

f

⁢

(

x

0

)

=

-

1

f

′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

.

In the case that the tangent is horizontal, the equation of the vertical normal is

x

=

x

0

,

and in the case that the tangent is vertical, the equation of the normal is

y

=

f

⁢

(

x

0

)

.

The normal of a curve at its point

P

always goes through the

center of curvature

belonging to the point

P

.

In the picture below, the black curve is a

parabola

, the red line is the tangent at the point

P

, and the blue line is the normal at the point

P

.

Title

normal line

Canonical name

NormalLine

Date of creation

2013-03-22 17:09:53

Last modified on

2013-03-22 17:09:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B05

Classification

msc 26A24

Classification

msc 53A04

Synonym

normal of curve

Synonym

normal

Synonym

perpendicular

Related topic

ConditionOfOrthogonality

Related topic

ParallelCurve

Related topic

SurfaceNormal

Related topic

Grafix

Related topic

NormalOfPlane

Defines

foot of normal

Defines

foot of perpendicular

Generated on Fri Feb 9 19:56:11 2018 by

LaTeXML
one-sided derivatives

one-sided derivatives

•

If the real function

f

is defined in the point

x

0

and on some

interval

left from this and if the left-hand

one-sided limit

lim

h

→

0

-

⁡

f

⁢

(

x

0

+

h

)

-

f

⁢

(

x

0

)

h

exists, then this limit is defined to be the

left-sided derivative

of

f

in

x

0

.

•

If the real function

f

is defined in the point

x

0

and on some interval right from this and if the right-hand one-sided limit

lim

h

→

0

+

⁡

f

⁢

(

x

0

+

h

)

-

f

⁢

(

x

0

)

h

exists, then this limit is defined to be the

right-sided derivative

of

f

in

x

0

.

It’s apparent that if

f

has both the left-sided and the right-sided derivative in the point

x

0

and these are equal, then

f

is

differentiable

in

x

0

and

f

′

⁢

(

x

0

)

equals to these one-sided derivatives.  Also inversely.

Example.

The real function

x

↦

x

⁢

x

is defined for

x

≧

0

and differentiable for

x

>

0

with

f

′

⁢

(

x

)

≡

3

2

⁢

x

.  The

function

also has the right derivative in

0

:

lim

h

→

0

+

⁡

h

⁢

h

-

0

⁢

0

h

=

lim

h

→

0

+

⁡

h

=

0

Remark.

For a function

f

:

[

a

,

b

]

→

ℝ

,
to have a right-sided derivative at

x

=

a

with value

d

,
is

equivalent

to saying that there is an

extension

g

of

f

to some open interval containing

[

a

,

b

]

and satisfying

g

′

⁢

(

a

)

=

d

.  Similarly for left-sided derivatives.

Title

one-sided derivatives

Canonical name

OnesidedDerivatives

Date of creation

2013-03-22 15:39:00

Last modified on

2013-03-22 15:39:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B05

Classification

msc 26A24

Synonym

left derivative

Synonym

right derivative

Related topic

Differentiable

Related topic

OneSidedLimit

Related topic

DifferntiableFunction

Related topic

OneSidedContinuity

Related topic

SemicubicalParabola

Defines

left-sided derivative

Defines

right-sided derivative

Generated on Fri Feb 9 19:56:13 2018 by

LaTeXML