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 $\hointr a b$.


This article, or a section of it, needs explaining.
In particular: Investigation needed as to whether there is a mistake in 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) -- should it actually be a closed interval?
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Let the series:

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

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


This article, or a section of it, needs explaining.
In particular: Is $f$ really defined at $x=b$? What are the domains of $\sequence {f_n}$?
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Then $f$ is continuous on $\hointr a b$.


Proof


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Sources

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