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