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[parent] limit function of sequence (Theorem)
Theorem 1   Let $ f_1, f_2, \ldots$ be a sequence of real functions all defined in the interval $ [a, b]$. This function sequence converges uniformly to the limit function $ f$ on the interval $ [a, b]$ if and only if
$\displaystyle \lim_{n\to\infty}\sup\{\vert f_n(x)-f(x)\vert\vdots a \leqq x \leqq b\} = 0.$

If all functions $ f_n$ are continuous in the interval $ [a, b]$ and $ \lim_{n\to\infty}f_n(x) = f(x)$ in all points $ x$ of the interval, the limit function needs not to be continuous in this interval; example $ f_n(x) = \sin^{n}x$ in $ [0, \pi]$:

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Theorem 2   If all the functions $ f_n$ are continuous and the sequence $ f_1, f_2, \ldots$ converges uniformly to a function $ f$ in the interval $ [a, b]$, then the limit function $ f$ is continuous in this interval.

Note. The notion of uniform convergence can be extended to the sequences of complex functions (the interval is replaced with some subset $ G$ of $ \mathbb{C}$). The limit function of a uniformly convergent sequence of continuous functions is continuous in $ G$.



"limit function of sequence" is owned by pahio. [ full author list (2) ]
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See Also: the limit of a uniformly convergent sequence of continuous functions is continuous

Also defines:  function sequence, limit function
Keywords:  continuity, uniform convergence

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Attachments:
sum function of series (Definition) by pahio
sets where sequence of continuous functions diverge (Derivation) by yotam
condition for uniform convergence of sequence of functions (Proof) by fernsanz
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Cross-references: uniformly convergent, subset, complex functions, points, continuous, functions, converges uniformly, interval, real functions, sequence
There are 7 references to this entry.

This is version 18 of limit function of sequence, born on 2004-09-23, modified 2006-10-02.
Object id is 6209, canonical name is LimitFunctionOfSequence.
Accessed 9105 times total.

Classification:
AMS MSC: 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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