Real Analysis/Uniform Convergence
|←Pointwise Convergence||Real Analysis
Definition: a sequence of real-valued functions fn(x) is uniformly convergent if there is a function f(x) such that for every ε>0 there is an N>0 such that when n>N for every x in the domain of the functions f, that |fn(x)-f(x)|<ε
Theorem (Uniform Convergence Theorem))
Let be a series of continuous functions that uniformly converges to a function . Then is continuous.
There exists an N such that for all n>N, for any x. Now let n>N, and consider the continuous function . Since it is continuous, there exists a such that if , then . Then so the function f(x) is continuous.