Editing Uniform convergence (section)

=== Generalizations ===

One may straightforwardly extend the concept to functions ''E'' → ''M'', where (''M'', ''d'') is a [[metric space]], by replacing <math>|f_n(x)-f(x)|</math> with <math>d(f_n(x),f(x))</math>.

The most general setting is the uniform convergence of [[net (mathematics)|net]]s of functions ''E'' → ''X'', where ''X'' is a [[uniform space]]. We say that the net <math>(f_\alpha)</math> ''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every [[entourage (topology)|entourage]] ''V'' in ''X'', there exists an <math>\alpha_0</math>, such that for every ''x'' in ''E'' and every <math>\alpha\geq \alpha_0</math>, <math>(f_\alpha(x),f(x))</math> is in ''V''. In this situation, uniform limit of continuous functions remains continuous.