Uniformly Convergent Series of Continuous Functions is Continuous

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Theorem

Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions.

Let each of $\left\langle{f_n}\right\rangle$ be continuous on the interval $\left[{a \,.\,.\, b}\right)$.



Let the series:

$\displaystyle f \left({x}\right) := \sum_{n \mathop = 1}^\infty f_n \left({x}\right)$

be uniformly convergent for all $x \in \left[{a \,.\,.\, b}\right]$.


Then $f$ is continuous on $\left[{a \,.\,.\, b}\right)$.


Proof


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