Editing Modulus of continuity (section)

===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.