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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
derivative of limit function diverges from limit of derivatives (Example) by pahio
integral of limit function (Theorem) by pahio
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Cross-references: uniformly convergent, subset, complex functions, points, continuous, functions, converges uniformly, interval, real functions, sequence
There are 9 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 13741 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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