Modulus of continuity

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In mathematics, the modulus of continuity is a precise way to measure the smoothness of a function. It is used as a delicate tool in mathematical analysis, to discuss highly non-smooth functions, which nonetheless enjoy some (very generalized) kind of smoothness. Similar but less refined notions such as Lipschitz continuity and Hölder class are subsumed by the explicit use of the modulus of continuity.

Above and below we use the word smooth in a free or intuitive sense, and not in the formal sense of a C^\infty function (for which see smooth function).

Sometimes instead of the term modulus of continuity a term of modulus of smoothness is used.

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[edit] Formal definition

Let f be some function with real value. Let t be some point and let δ be a positive number. We define the local modulus of continuity at the point t by

\left.\right.\omega_f(\delta;t)=\sup_{s:|s-t| \le \delta} |f(t)-f(s)|.

The modulus of continuity (sometimes called the global modulus of continuity) is defined by

\omega_f(\delta) = \sup_t \omega_f(\delta;t)

Notice that we never specified the domain of the function. One can think of an interval or the entire real line, but actually one can pick any metric space, though then one must change in the definition the expression s:|s-t|\le\delta to s:d(s,t)\le\delta where d is the distance in the space.

Intuitively, the smaller the modulus of continuity the smoother the function is.

[edit] Examples

We will mostly be interested in functions on a closed interval (or more generally, on a compact space), since for functions on the real line the connection between smoothness and the modulus of continuity is not obvious. For example, for the function f = sin(x2) it is easy to see that ωf(δ) = 2 for any δ, even though this is a smooth function. Therefore all the rest of the examples will be on a finite interval.

A differentiable function on an interval will satisfy

\omega_f(\delta)\leq C\delta

where C is some number that depends on the function (actually, it would be the maximum of the absolute value of the derivative). This is exactly the definition of a Lipschitz function, that is a function belongs to the Lipschitz class if its modulus of continuity decreases linearly. Notice that even on a finite closed interval the derivative may not be bounded, consider

f(x) = x^2 \sin (1/x^2)\,

for x ≠ 0 and

f(0)=0\,

on the interval [ − 1,1].


The Hölder classes also correspond to specific moduli of continuity. A function f belongs to the α-Hölder class if and only if

\omega_f(\delta)\leq C\delta^\alpha

for some number C (we assume here 0 < α < 1 to make this interesting).

For an example where it is useful to discuss a modulus of continuity even larger than δα, see Dini test.

[edit] History

Steffens (2006, p. 160) attributes the first usage of omega for the modulus of continuity to Lebesgue (1909, p. 309/p. 75) where omega refers to the oscillation of a Fourier transform. De la Vallee Poussin (1919, pp. 7-8) mentions both names (1) "modulus of continuity" and (2) "modulus of oscillation" and then concludes "but we choose (1) to draw attention to the usage we will make of it".

[edit] Modulus of continuity of higher orders

It can be seen that formal definition of the modulus uses notion of finite difference of first order:

\omega_f(\delta)=\omega(f, \delta)=\sup\limits_{x; |h|<\delta;}\left|\Delta_h(f,x)\right|.

If we replace that difference with a difference of order n we get a modulus of continuity of order n:

\omega_n(f, \delta)=\sup\limits_{x; |h|<\delta;}\left|\Delta^n_h(f,x)\right|.

[edit] References

  • Ch. de la Vallée Poussin, L'approximation des fonctions d'une variable réelle, Gauthier-Villars, Paris, 1952 (reprint of 1919 edition).
  • H. Lebesgue, Sur les intégrales singulières, Ann. Fac. Sci. Univ. Toulouse, ser 3 vol 1, 1909, 25-117, reproduced in: Henri Lebesgue, Œuvres scientifiques, Vol. 3., pp. 259-351.
  • K.-G. Steffens, The history of approximation theory, Birkhäuser, Boston 2006.
  • A.V. Efimov, Modulus of continuity, Encyclopaedia of Mathematics, Springer, 2001. ISBN 1402006098.