Editing Arzelà–Ascoli theorem (section)

===Non-continuous functions===
Solutions of numerical schemes for parabolic equations are usually piecewise constant, and therefore not continuous, in time. As their jumps nevertheless tend to become small as the time step goes to <math>0</math>, it is possible to establish uniform-in-time convergence properties using a generalisation to non-continuous functions of the classical Arzelà–Ascoli theorem (see e.g. {{harvtxt|Droniou|Eymard|2016|loc=Appendix}}).

Denote by <math>S(X,Y)</math> the space of functions from <math>X</math> to <math>Y</math> endowed with the uniform metric

:<math>d_S(v,w)=\sup_{t\in X}d_Y(v(t),w(t)).</math>

Then we have the following:

:Let <math>X</math> be a compact metric space and <math>Y</math> a complete metric space. Let <math>\{v_n\}_{n\in\mathbb{N}}</math> be a sequence in <math>S(X,Y)</math> such that there exists a function <math>\omega:X\times X\to[0,\infty]</math> and a sequence <math>\{\delta_n\}_{n\in\mathbb{N}}\subset[0,\infty)</math> satisfying
::<math>\lim_{d_X(t,t')\to0}\omega(t,t')=0,\quad\lim_{n\to\infty}\delta_n=0,</math>
::<math>\forall(t,t')\in X\times X,\quad \forall n\in\mathbb{N},\quad d_Y(v_n(t),v_n(t'))\leq \omega(t,t')+\delta_n.</math>
:Assume also that, for all <math>t\in X</math>, <math>\{v_n(t):n\in\mathbb{N}\}</math> is relatively compact in <math>Y</math>. Then <math>\{v_n\}_{n\in\mathbb{N}}</math> is relatively compact in <math>S(X,Y)</math>, and any limit of <math>\{v_n\}_{n\in\mathbb{N}}</math> in this space is in <math>C(X,Y)</math>.