limit function of sequence

Theorem 1.

Let  $f_1,f_2,\mathrm{\dots }$  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

$$\underset{n\to \mathrm{\infty }}{lim}sup\{|f_n(x)-f(x)|\mathrm{⋮}a\leqq x\leqq b\}=0.$$

If all functions $f_n$ are continuous in the interval  $[a,b]$  and  ${lim}_{n\to \mathrm{\infty }}{f}_n(x)=f(x)$  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  $f_1,f_2,\mathrm{\dots }$  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 $ℂ$).  The limit function of a uniformly convergent sequence of continuous functions is continuous in $G$.