Oscillation (mathematics)

Let ${\displaystyle (a_{n})}$  be a sequence of real numbers. The oscillation ${\displaystyle \omega (a_{n})}$  of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of ${\displaystyle (a_{n})}$ :

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

The oscillation is zero if and only if the sequence converges. It is undefined if ${\displaystyle \limsup _{n\to \infty }}$  and ${\displaystyle \liminf _{n\to \infty }}$  are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.

Let ${\displaystyle f}$  be a real-valued function of a real variable. The oscillation of ${\displaystyle f}$  on an interval ${\displaystyle I}$  in its domain is the difference between the supremum and infimum of ${\displaystyle f}$ :

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

More generally, if ${\displaystyle f:X\to \mathbb {R} }$  is a function on a topological space ${\displaystyle X}$  (such as a metric space), then the oscillation of ${\displaystyle f}$  on an open set ${\displaystyle U}$  is

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

The oscillation of a function ${\displaystyle f}$  of a real variable at a point ${\displaystyle x_{0}}$  is defined as the limit as ${\displaystyle \epsilon \to 0}$  of the oscillation of ${\displaystyle f}$  on an ${\displaystyle \epsilon }$ -neighborhood of ${\displaystyle x_{0}}$ :

${\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).}$ 

This is the same as the difference between the limit superior and limit inferior of the function at ${\displaystyle x_{0}}$ , provided the point ${\displaystyle x_{0}}$  is not excluded from the limits.

More generally, if ${\displaystyle f:X\to \mathbb {R} }$  is a real-valued function on a metric space, then the oscillation is

${\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).}$ 

 

• ${\displaystyle {\frac {1}{x}}}$  has oscillation ∞ at ${\displaystyle x}$  = 0, and oscillation 0 at other finite ${\displaystyle x}$  and at −∞ and +∞.
• ${\displaystyle \sin {\frac {1}{x}}}$  (the topologist's sine curve) has oscillation 2 at ${\displaystyle x}$  = 0, and 0 elsewhere.
• ${\displaystyle \sin x}$  has oscillation 0 at every finite ${\displaystyle x}$ , and 2 at −∞ and +∞.
• ${\displaystyle (-1)^{x}}$ or 1, −1, 1, −1, 1, −1... has oscillation 2.

In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.

Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.

Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x₀ if and only if the oscillation is zero;^[1] in symbols, ${\displaystyle \omega _{f}(x_{0})=0.}$  A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

For example, in the classification of discontinuities:

• in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
• in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
• in an essential discontinuity, oscillation measures the failure of a limit to exist.

This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a G_δ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.^[2]

The oscillation is equivalent to the ε-δ definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε₀ there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε₀, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

More generally, if f : X → Y is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each x ∈ X by

${\displaystyle \omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}}$ 

• Wave equation
• Wave envelope
• Grandi's series
• Bounded mean oscillation