Editing Continuous function (section)

====Definition in terms of control of the remainder====
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. 
A function <math>C: [0,\infty) \to [0,\infty]</math> is called a control function if
* ''C'' is non-decreasing
*<math>\inf_{\delta > 0} C(\delta) = 0</math>

A function <math>f : D \to R</math> is ''C''-continuous at <math>x_0</math> if there exists such a neighbourhood <math display="inline">N(x_0)</math> that 
<math display="block">|f(x) - f(x_0)| \leq C\left(\left|x - x_0\right|\right) \text{ for all } x \in D \cap N(x_0)</math>

A function is continuous in <math>x_0</math> if it is ''C''-continuous for some control function ''C''.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions <math>\mathcal{C}</math> a function is {{nowrap|<math>\mathcal{C}</math>-continuous}} if it is {{nowrap|<math>C</math>-continuous}} for some <math>C \in \mathcal{C}.</math> For example, the [[Lipschitz continuity|Lipschitz]], the [[Hölder continuous function]]s of exponent {{mvar|α}} and the [[uniformly continuous function]]s below are defined by the set of control functions 
<math display="block">\mathcal{C}_{\mathrm{Lipschitz}} = \{C : C(\delta) = K|\delta| ,\  K > 0\}</math> 
<math display="block">\mathcal{C}_{\text{Hölder}-\alpha} = \{C : C(\delta) = K |\delta|^\alpha, \ K > 0\}</math>
<math display="block">\mathcal{C}_{\text{uniform cont.}} = \{C : C(0) = 0 \}</math>
respectively.