Editing Modulus of continuity (section)

==Special moduli of continuity==
Special moduli of continuity also reflect certain global properties of functions such as extendibility and uniform approximation. In this section we mainly deal with moduli of continuity that are [[concave function|concave]], or [[subadditive]], or uniformly continuous, or sublinear. These properties are essentially equivalent in that, for a modulus ω (more precisely, its restriction on [0, ∞)) each of the following implies the next:
*ω is concave;
*ω is subadditive;
*ω is uniformly continuous;
*ω is sublinear, that is, there are constants ''a'' and ''b'' such that ω(''t'') ≤ ''at''+''b'' for all ''t'';
*ω is dominated by a concave modulus, that is, there exists a concave modulus of continuity <math>\tilde\omega</math> such that <math>\omega(t)\leq \tilde\omega(t)</math> for all ''t''.

Thus, for a function ''f'' between metric spaces it is equivalent to admit a modulus of continuity which is either concave, or subadditive, or uniformly continuous, or sublinear. In this case, the function ''f'' is sometimes called a ''special uniformly continuous'' map. This is always true in case of either compact or convex domains. Indeed, a uniformly continuous map ''f'' : ''C'' → ''Y'' defined on a [[convex set]] ''C'' of a normed space ''E'' always admits a [[subadditive]] modulus of continuity; in particular, real-valued as a function ω : [0, ∞) → [0, ∞). Indeed, it is immediate to check that the optimal modulus of continuity ω<sub>''f''</sub> defined above is subadditive if the domain of ''f'' is convex: we have, for all ''s'' and ''t'':

:<math>\begin{align}
\omega_f(s+t) &=\sup_{|x-x'|\le t+s} d_Y(f(x),f(x')) \\
&\leq \sup_{|x-x'|\le t+s}\left\{d_Y\left( f(x), f\left(x-t\frac{x-x'}{|x-x'|}\right)\right) + d_Y\left( f\left(x-t\frac{x-x'}{|x-x'|}\right), f(x')\right )\right\} \\
&\leq \omega_f(t)+\omega_f(s).
\end{align}</math>

Note that as an immediate consequence, any uniformly continuous function on a convex subset of a normed space has a sublinear growth: there are constants ''a'' and ''b'' such that |''f''(''x'')| ≤ ''a''|''x''|+''b'' for all ''x''. However, a uniformly continuous function on a general metric space admits a concave modulus of continuity if and only if the ratios <math>d_Y(f(x),f(x'))/d_X(x,x')</math> are uniformly bounded for all pairs (''x'', ''x''′) with distance bounded away from zero; this condition is certainly satisfied by any bounded uniformly continuous function; hence in particular, by any continuous function on a compact metric space.

===Sublinear moduli, and bounded perturbations from Lipschitz===
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if ''f'' is a uniformly continuous function with modulus of continuity ω, and ''g'' is a ''k'' Lipschitz function with uniform distance ''r'' from ''f'', then ''f'' admits the sublinear modulus of continuity min{ω(''t''), 2''r''+''kt''}. Conversely, at least for real-valued functions, any special uniformly continuous function is a bounded, uniformly continuous perturbation of some Lipschitz function; indeed more is true as shown below (Lipschitz approximation).

===Subadditive moduli, and extendibility===
The above property for uniformly continuous function on convex domains admits a sort of converse at least in the case of real-valued functions: that is, every special uniformly continuous real-valued function ''f'' : ''X'' → '''R''' defined on a metric space ''X'', which is a metric subspace of a normed space ''E'', admits extensions over ''E'' that preserves any subadditive modulus ω of ''f''. The least and the greatest of such extensions are respectively:

:<math>\begin{align}
f_*(x) &:=\sup_{y\in X}\left\{f(y)-\omega(|x-y|)\right\}, \\
f^*(x) &:=\inf_{y\in X}\left\{f(y)+\omega(|x-y|)\right\}.
\end{align}</math>

As remarked, any subadditive modulus of continuity is uniformly continuous: in fact, it admits itself as a modulus of continuity. Therefore, ''f''<sub>∗</sub> and ''f*'' are respectively inferior and superior envelopes of ω-continuous families; hence still ω-continuous. Incidentally, by the [[Kuratowski embedding]] any metric space is isometric to a subset of a normed space. Hence, special uniformly continuous real-valued functions are essentially the restrictions of uniformly continuous functions on normed spaces. In particular, this construction provides a quick proof of the [[Tietze extension theorem]] on compact metric spaces. However, for mappings with values in more general Banach spaces than '''R''', the situation is quite more complicated; the first non-trivial result in this direction is the [[Kirszbraun theorem]].

===Concave moduli and Lipschitz approximation===
Every special uniformly continuous real-valued function ''f'' : ''X'' → '''R''' defined on the metric space ''X'' is [[uniform convergence|uniformly]] approximable by means of Lipschitz functions. Moreover, the speed of convergence in terms of the Lipschitz constants of the approximations is strictly related to the modulus of continuity of ''f''. Precisely, let ω be the minimal concave modulus of continuity of ''f'', which is
:<math>\omega(t)=\inf\big\{at+b\, :\, a>0,\, b>0,\, \forall x\in X,\, \forall x'\in X\,\,  |f(x)-f(x')|\leq ad(x,x')+b\big\}.</math>
Let δ(''s'') be the uniform [[metric spaces#Distance between points and sets; Hausdorff distance and Gromov metric|distance]] between the function ''f'' and the set Lip<sub>''s''</sub> of all Lipschitz real-valued functions on ''C'' having Lipschitz constant ''s'' :
:<math>\delta(s):=\inf\big\{\|f-u\|_{\infty,X}\,:\, u\in \mathrm{Lip}_s\big\}\leq+\infty.</math>
Then the functions ω(''t'') and δ(''s'') can be related with each other via a [[Legendre transformation]]: more precisely, the functions 2δ(''s'') and −ω(−''t'') (suitably extended to +∞ outside their domains of finiteness) are a pair of conjugated convex functions,<ref>[https://mathoverflow.net/q/194890 Legendre transform and Lipschitz approximation]</ref> for

:<math>2\delta(s)=\sup_{t\geq0}\left\{\omega(t)-st\right\},</math>
:<math>\omega(t)=\inf_{s\geq0}\left\{2\delta(s)+st\right\}.</math>

Since ω(''t'') = o(1) for ''t'' → 0<sup>+</sup>, it follows that δ(''s'') = o(1) for ''s'' → +∞, that exactly means that ''f'' is uniformly approximable by Lipschitz functions. Correspondingly, an optimal approximation is given by the functions
:<math>f_s:=\delta(s)+\inf_{y\in X}\{f(y)+sd(x,y)\}, \quad  \mathrm{for} \ s\in\mathrm{dom}(\delta):</math>
each function ''f<sub>s</sub>'' has Lipschitz constant ''s'' and
:<math>\|f-f_s\|_{\infty,X}=\delta(s);</math>
in fact, it is the greatest ''s''-Lipschitz function that realize the distance δ(''s''). For example, the α-Hölder real-valued functions on a metric space are characterized as those functions that can be uniformly approximated by ''s''-Lipschitz functions with speed of convergence <math>O(s^{-\frac{\alpha}{1-\alpha}}),</math> while the almost Lipschitz functions are characterized by an exponential speed of convergence <math>O(e^{-as}).</math>