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:

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

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:

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

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.

$\blacksquare$