Continuity, modulus of

2020 Mathematics Subject Classification: Primary: 54C05 [MSN][ZBL]

One of the basic characteristics of (uniformly) continuous functions. The modulus of a continuity of a continuous function $f:\mathbb R^n \supset E \to \mathbb R^k$ is given by \[ \omega (\delta, f) := \sup_{|x-y|\leq \delta} |f(x)-f(y)|\, , \] where we are implicitly assuming that $\lim_{\delta\downarrow 0} \omega (\delta, f) = 0$, which is indeed true if and only if $f$ is uniformly continuous. The definition of the modulus of continuity was introduced by H. Lebesgue in 1910 for functions of one real variable, although in essence the concept was known earlier. The definition can be readily extended to uniformly continuous maps $f$ between metric spaces $(X,d_X)$ and $(Y, d_Y)$, simply setting \[ \omega (\delta, f) := \sup_{d_X (x,y)\leq \delta} d_Y (f(x), f(y))\, . \]

Examples

Very special classes of moduli of continuity give notable classes of functions. For instance

• If $\omega (\delta, f) \leq M \delta$ for some constant $M>0$, then $f$ satisfies the Lipschitz condition; the least constant $M$ for which such inequality holds is the Lipschitz constant of $f$; cf. also Lipschitz function.
• If $\omega (\delta, f) \leq M \delta^\alpha$ for some constants $M>0$, $\alpha\in ]0,1]$, then $f$ satisfies the Hölder condition of exponent $\alpha$.
• If

\[ \int_0^1 \frac{\omega (\delta, f)}{\delta}\, d\delta < \infty\, , \] then $f$ is called, by some authors, Dini continuous (such condition plays a special role in the convergence of Fourier series, cf. Dini criterion).

Basic properties

It is elementary to derive bounds on the modulus of continuity of linear combinations, compositions and infima of uniformly continuous functions in term of their respective moduli of continuity. In particular

• $\omega (\delta, \lambda f + \mu g) \leq |\lambda| \omega (\delta, f) + |\mu| \omega (\delta, g)$;
• $\omega (\delta, g\circ f) \leq \omega (\omega (\delta, f), g)$;
• If $\{f_\lambda\}$ is a family of real-valued functions with $\omega (\delta, f_\lambda) \leq h (\delta)$ for some common function $h$, then $\inf_\lambda f_\lambda$ and $\sup_\lambda f_\lambda$ are also continuous with modulus of continuity bounded by $h$, provided the respective infima and suprema are finite (for which it is indeed necessary and sufficient that they are finite at some point).

For a non-negative function $\omega: [0,\infty[\to \mathbb R$ there is a continuous function $f: [0, \infty[ \to \mathbb R$ with $\omega (\delta, f) = \omega (\delta)$ for every $\delta$ if and only if the following properties hold:

• $\omega (0) = 0$
• $\omega$ is continuous
• $\omega$ is a subadditive function, namely $\omega (\alpha + \beta) \leq \omega (\alpha) + \omega (\beta)$ for every $\alpha, \beta \geq 0$.

Generalizations

As already mentioned, the notion can be easily generalized to maps between arbitrary metric spaces.

One can also consider moduli of continuity of higher orders. For instance, if $f$ is a function of one variable, the modulus of continuity can be rewritten as \[ \omega (\delta, f) = \sup_{|h| \leq \delta} \max_x \Delta_h f (x)\, , \] where $\Delta_h f (x) = |f(x+h)-f(x)|$ is the usual finite difference of first order. Therefore, if we introduce the higher oder finite differences \[ \Delta^k_h (x) = \sum_{i =0}^k (-1)^{k-i} {{k}\choose{i}} f (x+ih)\, , \] the higher oder moduli of continuity can be defined as \[ \omega_k (\delta, f) = \sup_{|h|\leq \delta} \max_x \Delta^k_h f (x)\, . \] See also Smoothness, modulus of. Moduli of continuity and smoothness are extensively used in approximation theory and Fourier analysis (cf. Harmonic analysis).

A further common generalization replaces pointwise maxima with integrals. For instance, if $f: \mathbb R^n \to \mathbb R$ is Lebesgue measurable, the $L^p$-modulus of continuity of $f$ is defined as \[ \omega^{(p)} (\delta, f) := \sup_{|\xi|\leq \delta} \int |f(x+\xi) - f(x)|^p\, dx\, . \] An obvious variant can be defined for maps on open domains $\Omega$ by simply restricting the domain of integration to $\{x\in \Omega : {\rm dist}\, (x, \partial \Omega) < \delta\}$. A classical characterization of the Sobolev space $W^{1,p} (\mathbb R^n)$ is then the following

Theorem Let $p \in ]1, \infty[$. $f\in L^p (\mathbb R^n)$ belongs to $W^{1,p}$ if and only if there is a constant $M$ such that $\omega^{(p)} (\delta, f) \leq M \delta^p$ for every $\delta$.

Cf. Theorem 3 in Section 5.8 of [Ev]. The limiting case $p=\infty$ of the above theorem is also valid and gives then the identity between $W^{1,\infty} (\mathbb R^n)$ and the space ${\rm Lip}_b (\mathbb R^n)$ of bounded Lipschitz functions. For $p=1$ the property $\omega^{(1)} (\delta, f) \leq M \delta$ characterizes instead the space of functions of bounded variation.

References

[AGS]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Dz]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[Ev]	L.C. Evans, "Partial differential equations", Graduate studies in mathematics. American Mathematical Society (1998).
[St]	K. G. Steffens, "The History of Approximation Theory", Birkhäuser (2006).
[Zy]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988).