Uniformly Convergent Series of Continuous Functions Converges to Continuous Function
Theorem
Let $S \subseteq \R$.
Let $x \in S$.
Let $\sequence {f_n}$ be a sequence of real functions.
Let $f_n$ be continuous at $x$ 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 at $x$.
Corollary
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 $\sequence {s_n}$ be sequence of real functions $S \to \R$ such that:
- $\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$
for each $n \in \N$ and $x \in S$.
By Sum Rule for Continuous Real Functions:
- $s_n$ is continuous at $x$ for all $n \in \N$.
Since additionally $s_n \to f$ uniformly, we have by Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function:
- $f$ is continuous at $x$.
$\blacksquare$
Sources
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.6$: Uniform Convergence of Infinite Series of Functions: Theorem $9.7$