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Homelimit function of sequence

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# 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 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 $G$.

## Mathematics Subject Classification

26A15*no label found*40A30

*no label found*

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