If fn is a sequence of continuous functions converging uniformly to a function f, then f is also continuous.
If fn is a sequence of Riemann integrable functions of an interval [a,b] converging uniformly to a function f, then f is also Riemann integrable and ∫fn converges to ∫f.
If fn is a sequence of holomorphic functions converging uniformly to a function f, then f is holomorphic.
(Uniform convergence on compacta) If fn is a sequence of holomorphic functions converging to f, and if the convergence is uniform over any compact set, then f is holomorphic.