Editing Limit of a function (section)

===Uniform metric===

Finally, we will discuss the limit in [[function space]], which has infinite dimensions. Consider a function {{math|''f''(''x'', ''y'')}} in the function space <math>S \times T \to \R.</math> We want to find out as {{mvar|x}} approaches {{mvar|p}}, how {{math|''f''(''x'', ''y'')}} will tend to another function {{math|''g''(''y'')}}, which is in the function space <math>T \to \R.</math> The "closeness" in this function space may be measured under the [[uniform metric]].<ref>{{citation
 | last = Rudin | first = W
 | url = http://worldcat.org/oclc/962920758
 | title = Principles of mathematical analysis
 | date = 1986
 | publisher = McGraw - Hill Book C
 | pages = 150–151
 | oclc = 962920758}}</ref> Then, we will say '''the uniform limit of {{mvar|f}} on {{mvar|T}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|g}}''' and write
<math display=block>\underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y),</math> or
<math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T,</math>

if the following holds:

{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{mvar|x}} in {{mvar|S}}, {{math|0 < {{!}}''x'' − ''p''{{!}} < ''δ''}} implies <math>\sup_{y \in T}|f(x,y) - g(y)| < \varepsilon.</math>}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S) \,(0 < |x-p| < \delta \implies \sup_{y \in T} | f(x, y) - g(y) | < \varepsilon).</math>

In fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.