limit function of sequence

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

 $\lim_{n\to\infty}\sup\{|f_{n}(x)-f(x)|\vdots\,\,a\leqq x\leqq b\}=0.$

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  $[0,\,\pi]$:

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 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 $G$.

Title limit function of sequence LimitFunctionOfSequence 2013-03-22 14:37:45 2013-03-22 14:37:45 pahio (2872) pahio (2872) 22 pahio (2872) Theorem msc 26A15 msc 40A30 LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous PointPreventingUniformConvergence function sequence limit function