modulus of continuity

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 Hlder 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).

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)=\max_{s:|s-t| < \delta} |f(t)-f(s)|.

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

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

Notice that we never defined 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|<\delta

into

s:d(s,t)<\delta

where d is the distance in the space. Intuitively, the smaller the modulus of continuity the smoother the function is.

Examples

We will mostly be interested in functions on an 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

f=\sin(x^2)

it is easy to see that

\omega_f(\delta)=2

for any δ, even though this is a nice smooth function. Therefore all the rest of the examples will be on an 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 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. The Hlder classes are also a private case of a modulus of continuity inequality. A function f belongs to the α-Hlder class if and only if

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

for some number C (we assume here

0<\alpha<1

to make this interesting). For an example where it is useful to discuss a modulus of continuity even larger than

\delta^\alpha

, see Dini test.

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