# semi-continuous

A real function $f:A\to \mathbb{R}$, where $A\subseteq \mathbb{R}$ is said to be *lower semi-continuous* in ${x}_{0}$ if

$$ |

and $f$ is said to be *upper semi-continuous* if

$$ |

## Remark

A real function is continuous^{} in ${x}_{0}$ if and only if it is both upper and lower semicontinuous in ${x}_{0}$.

We can generalize the definition to arbitrary topological spaces^{} as follows.

Let $A$ be a topological space.
$f:A\to \mathbb{R}$ is lower semicontinuous at ${x}_{0}$ if, for each $\epsilon >0$ there is a neighborhood^{} $U$ of ${x}_{0}$ such that $x\in U$ implies $f(x)>f({x}_{0})-\epsilon $.

## Theorem

Let $f:[a,b]\to \mathbb{R}$ be a lower (upper) semi-continuous function. Then $f$ has a minimum (maximum) in $[a,b]$.

Title | semi-continuous |
---|---|

Canonical name | Semicontinuous |

Date of creation | 2013-03-22 12:45:41 |

Last modified on | 2013-03-22 12:45:41 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 26A15 |

Classification | msc 54-XX |

Synonym | semicontinuous |