# Uniformly Convergent Series of Continuous Functions is Continuous

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## Theorem

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

Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$.

Let the series:

- $\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$

be uniformly convergent for all $x \in \closedint a b$.

Then $f$ is continuous on $\hointr a b$.

## Proof

## Sources

- 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.3.1$: Properties of uniformly convergent series: Theorem $1.9 \ \text{(a)}$