quality:correction, linguistic:translation
Generalizations

One may straightforwardly extend the concept to functions E → M, where (M, d) is a metric space, by replacing | f n (x) − f (x) | {\displaystyle |f_{n}(x)-f(x)|} with d (f n (x), f (x)) {\displaystyle d(f_{n}(x),f(x))}.

The most general setting is the uniform convergence of nets of functions E → X, where X is a uniform space. We say that the net (f α) {\displaystyle (f_{\alpha })} converges uniformly with limit f : E → X if and only if for every entourage V in X, there exists an α 0 {\displaystyle \alpha _{0}}, such that for every x in E and every α ≥ α 0 {\displaystyle \alpha \geq \alpha _{0}}, (f α (x), f (x)) {\displaystyle (f_{\alpha }(x),f(x))} is in V. In this situation, uniform limit of continuous functions remains continuous.