# 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

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

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 ${f}_{n}(x)={\mathrm{sin}}^{n}x$ 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 $\u2102$). The limit function of a uniformly convergent sequence of continuous functions is continuous in $G$.

Title | limit function of sequence |
---|---|

Canonical name | LimitFunctionOfSequence |

Date of creation | 2013-03-22 14:37:45 |

Last modified on | 2013-03-22 14:37:45 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 22 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26A15 |

Classification | msc 40A30 |

Related topic | LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous |

Related topic | PointPreventingUniformConvergence |

Defines | function sequence |

Defines | limit function |