limit function of sequenceLimitFunctionOfSequence2004-09-23 17:50:262006-10-02 11:17:38Theoremuniform convergencefunction sequencelimit functioncontinuityuniform convergence\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem{thmplain}{Theorem}\begin{thmplain}
\, Let\, $f_1,\,f_2,\,\ldots$\, be a sequence of real functions all defined in the interval\, $[a,\,b]$.\, This {\em function sequence} converges uniformly to the {\em limit function} $f$ on the interval\, $[a,\,b]$\, if and only if
$$\lim_{n\to\infty}\sup\{|f_n(x)-f(x)|\vdots \,\, a \leqq x \leqq b\} = 0.$$
\end{thmplain}
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\, $[0,\,\pi]$:
\begin{center}
\includegraphics{pmplot}
\end{center}
\begin{thmplain}
\, 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.
\end{thmplain}
\textbf{Note.} \,The notion of \PMlinkescapetext{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 $G$.