Oscillation (mathematics)

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set).

Definitions

Oscillation of a sequence

Let

(

a

n

)

{\displaystyle (a_{n})}

be a sequence of real numbers. The oscillation

ω

(

a

n

)

{\displaystyle \omega (a_{n})}

of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of

(

a

n

)

{\displaystyle (a_{n})}

:

ω

(

a

n

)

=

lim sup

n

→

∞

a

n

−

lim inf

n

→

∞

a

n

{\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}}

.

The oscillation is zero if and only if the sequence converges. It is undefined if

lim sup

n

→

∞

{\displaystyle \limsup _{n\to \infty }}

and

lim inf

n

→

∞

{\displaystyle \liminf _{n\to \infty }}

are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.

Oscillation of a function on an open set

Let

f

{\displaystyle f}

be a real-valued function of a real variable. The oscillation of

f

{\displaystyle f}

on an interval

I

{\displaystyle I}

in its domain is the difference between the supremum and infimum of

f

{\displaystyle f}

:

ω

f

(

I

)

=

sup

x

∈

I

f

(

x

)

−

inf

x

∈

I

f

(

x

)

.

{\displaystyle \omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).}

More generally, if

f

:

X

→

R

{\displaystyle f:X\to \mathbb {R} }

is a function on a topological space

X

{\displaystyle X}

(such as a metric space), then the oscillation of

f

{\displaystyle f}

on an open set

U

{\displaystyle U}

is

ω

f

(

U

)

=

sup

x

∈

U

f

(

x

)

−

inf

x

∈

U

f

(

x

)

.

{\displaystyle \omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).}

Oscillation of a function at a point

The oscillation of a function

f

{\displaystyle f}

of a real variable at a point

x

0

{\displaystyle x_{0}}

is defined as the limit as

ϵ

→

0

{\displaystyle \epsilon \to 0}

of the oscillation of

f

{\displaystyle f}

on an

ϵ

{\displaystyle \epsilon }

-neighborhood of

x

0

{\displaystyle x_{0}}

:

ω

f

(

x

0

)

=

lim

ϵ

→

0

ω

f

(

x

0

−

ϵ

,

x

0

+

ϵ

)

.