Continuity of the Sum of a Uniformly Convergent Series of Functions

# Continuity of the Sum of a Uniformly Convergent Series of Functions

Recall from the Pointwise Convergent and Uniformly Convergent Series of Functions that we say that the series of functions $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is uniformly convergent to the sum function $f(x)$ if the corresponding sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ uniformly converges to $f(x)$ where for each $n \in \mathbb{N}$ we define $\displaystyle{s_n(x) = \sum_{k=1}^{n} f_n(x)}$.

Now suppose that we have a sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$, each of which is continuous at the point $c \in X$. Suppose further that $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to a sum function $f(x)$. Is the sum function $f$ necessarily continuous at $c$? The answer is yes as we will prove in the following theorem.

 Theorem 1: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of functions with common domain $X$ and suppose that $f_n$ is continuous at $c \in X$ for all $n \in \mathbb{N}$. If $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to the sum function $f(x)$ then $f$ is continuous at $c$.
• Proof: Suppose that $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to the sum function $f(x)$. Let $(s_n(x))_{n=1}^{\infty}$ be the corresponding sequence of partial sums to this series of functions. Then $(s_n(x))_{n=1}^{\infty}$ converges uniformly to $f(x)$.
• Now for each $n \in \mathbb{N}$ we have that:
(1)
\begin{align} \quad s_n(x) = \sum_{k=1}^{n} f_n(x) = f_1(x) + f_2(x) + ... + f_n(x) \end{align}