Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary

Theorem
Let $S \subseteq \R$.

Let $\sequence {f_n}$ be a sequence of real functions.

Let $f_n$ be continuous for all $n \in \N$.

Let the infinite series:


 * $\displaystyle \sum_{n \mathop = 1}^\infty f_n$

be uniformly convergent to a real function $f : S \to \R$.

Then $f$ is continuous.

Proof
Let $x \in S$.

Then $f_n$ is continuous at $x$ for all $n \in \N$.

Since:


 * $\displaystyle \sum_{n \mathop = 1}^\infty f_n$

converges uniformly to $f$, we have by Uniformly Convergent Series of Continuous Functions Converges to Continuous Function:


 * $f$ is continuous at $x$.

As $x \in S$ was arbitrary, we have that:


 * $f$ is continuous.