Editing Modulus of continuity (section)

===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>