Editing Pointwise convergence (section)

==Definition==
[[File:Drini-nonuniformconvergence.png|thumb|300px|
The pointwise limit of continuous functions does not have to be continuous: the continuous functions 
<math>\sin^n(x)</math> (marked in green) converge pointwise to a discontinuous function (marked in red).]]
Suppose that <math>X</math> is a set and <math>Y</math> is a [[topological space]], such as the [[Real number|real]] or [[complex numbers]] or a [[metric space]], for example. A [[sequence]] of [[Function (mathematics)|functions]] <math>\left(f_n\right)</math> all having the same domain <math>X</math> and [[codomain]] <math>Y</math> is said to '''converge pointwise''' to a given function <math>f : X \to Y</math> often written as
<math display=block>\lim_{n\to\infty} f_n = f\  \mbox{pointwise}</math>
if (and only if) the [[limit of a sequence|limit of the sequence]] <math>f_n(x)</math> evaluated at each point <math>x</math> in the domain of <math>f</math> is equal to <math>f(x)</math>, written as
<math display=block>\forall x \in X, \lim_{n\to\infty} f_n(x) = f(x).</math>
The function <math>f</math> is said to be the '''pointwise limit''' function of the <math>\left(f_n\right).</math>

The definition easily generalizes from sequences to [[Net (mathematics)|net]]s <math>f_\bull = \left(f_a\right)_{a \in A}</math>. We say <math>f_\bull</math> converges pointwise to <math>f</math>, written as
<math display=block>\lim_{a\in A} f_a = f\  \mbox{pointwise}</math>
if (and only if) <math>f(x)</math> is the unique accumulation point of the net <math>f_\bull(x)</math> evaluated at each point <math>x</math> in the domain of <math>f</math>, written as
<math display=block>\forall x \in X, \lim_{a\in A} f_a(x) = f(x).</math>

Sometimes, authors use the term '''bounded pointwise convergence''' when there is a constant  <math>C</math> such that  <math>\forall n,x,\;|f_n(x)|<C</math>  .<ref>{{Cite book |last=Li |first=Zenghu |title=Measure-Valued Branching Markov Processes |publisher=Springer |year=2011 |isbn=978-3-642-15003-6}}</ref>