Editing Modulus of continuity (section)

==Formal definition==
Formally, a modulus of continuity is any increasing real-extended valued function ω : [0, ∞] → [0, ∞], vanishing at 0 and continuous at 0, that is
:<math>\lim_{t\to0}\omega(t)=\omega(0)=0.</math>

Moduli of continuity are mainly used to give a quantitative account both of the continuity at a point, and of the uniform continuity, for functions between metric spaces, according to the following definitions.

A function ''f'' : (''X'', ''d<sub>X</sub>'') → (''Y'', ''d<sub>Y</sub>'') admits ω as (local) modulus of continuity at the point ''x'' in ''X'' if and only if,
:<math>\forall x'\in X: d_Y(f(x),f(x'))\leq\omega(d_X(x,x')).</math>
Also, ''f'' admits ω as (global) modulus of continuity if and only if,
:<math>\forall x,x'\in X: d_Y(f(x),f(x'))\leq\omega(d_X(x,x')).</math>
One equivalently says that ω is a modulus of continuity (resp., at ''x'') for ''f'', or shortly, ''f'' is ω-continuous (resp., at ''x''). Here, we mainly treat the global notion.

===Elementary facts===
*If ''f'' has ω as modulus of continuity and ω<sub>1</sub> ≥ ω, then ''f'' admits ω<sub>1</sub> too as modulus of continuity.
*If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are functions between metric spaces with moduli respectively ω<sub>1</sub> and ω<sub>2</sub> then the composition map <math>g\circ f:X\to Z</math> has modulus of continuity <math>\omega_2\circ\omega_1</math>.
*If ''f'' and ''g'' are functions from the metric space X to the Banach space ''Y'', with moduli respectively ω<sub>1</sub> and ω<sub>2</sub>, then any linear combination ''af''+''bg'' has modulus of continuity |''a''|ω<sub>1</sub>+|''b''|ω<sub>2</sub>. In particular, the set of all functions from ''X'' to ''Y'' that have ω as a modulus of continuity is a convex subset of the vector space ''C''(''X'', ''Y''), closed under [[pointwise convergence]].
*If ''f'' and ''g'' are bounded real-valued functions on the metric space ''X'', with moduli respectively ω<sub>1</sub> and ω<sub>2</sub>, then the pointwise product ''fg'' has modulus of continuity <math>\|g\|_\infty\omega_1+\|f\|_\infty \omega_2</math>.
*If <math>\{f_\lambda\}_{\lambda\in\Lambda}</math> is a family of real-valued functions on the metric space ''X'' with common modulus of continuity ω, then the inferior envelope <math>\inf_{\lambda\in\Lambda}f_\lambda</math>, respectively, the superior envelope <math>\sup_{\lambda\in\Lambda}f_\lambda</math>, is a real-valued function with modulus of continuity ω, provided it is finite valued at every point. If ω is real-valued, it is sufficient that the envelope be finite at one point of ''X'' at least.

===Remarks===
*Some authors do not require monotonicity, and some require additional properties such as ω being continuous. However, if f admits a modulus of continuity in the weaker definition, it also admits a modulus of continuity which is increasing and infinitely differentiable in (0, ∞). For instance, <math display="block">\omega_1(t) := \sup_{s\leq t}\omega(s)</math> is increasing, and ω<sub>1</sub> ≥ ω; <math display="block">\omega_2(t):=\frac{1}{t} \int_t^{2t}\omega_1(s)ds</math> is also continuous, and ω<sub>2</sub> ≥ ω<sub>1</sub>, <br/> and a suitable variant of the preceding definition also makes ω<sub>2</sub> infinitely differentiable in [0, ∞].
*Any uniformly continuous function admits a minimal modulus of continuity ω<sub>''f''</sub>, that is sometimes referred to as ''the'' (optimal) modulus of continuity of ''f'': <math display="block">\omega_f(t) := \sup\{ d_Y(f(x),f(x')):x\in X,x'\in X,d_X(x,x')\le t \} ,\quad\forall t\geq0.</math> Similarly, any function continuous at the point ''x'' admits a minimal modulus of continuity at ''x'', ω<sub>''f''</sub>(''t''; ''x'') (''the'' (optimal) modulus of continuity of ''f'' at ''x'') : <math display="block">\omega_f(t;x):=\sup\{ d_Y(f(x),f(x')): x'\in X,d_X(x,x')\le t \},\quad\forall t\geq0.</math> However, these restricted notions are not as relevant, for in most cases the optimal modulus of ''f'' could not be computed explicitly, but only bounded from above (by ''any'' modulus of continuity of ''f''). Moreover, the main properties of moduli of continuity concern directly the unrestricted definition.
*In general, the modulus of continuity of a uniformly continuous function on a metric space needs to take the value +∞. For instance, the function ''f'' : '''N''' → '''R''' such that ''f''(''n'') := ''n''<sup>2</sup> is uniformly continuous with respect to the [[discrete metric]] on '''N''', and its minimal modulus of continuity is ω<sub>''f''</sub>(''t'') = +∞ for any  ''t''≥1, and ω<sub>''f''</sub>(''t'') = 0 otherwise. However, the situation is different for uniformly continuous functions defined on compact or convex subsets of normed spaces.