# one-sided continuity

The real function $f$ is continuous^{} from the left in the point $x={x}_{0}$ iff

$$\underset{x\to {x}_{0}-}{lim}f(x)=f({x}_{0}).$$ |

The real function $f$ is continuous from the right in the point $x={x}_{0}$ iff

$$\underset{x\to {x}_{0}+}{lim}f(x)=f({x}_{0}).$$ |

The real function $f$ is continuous on the closed interval^{} $[a,b]$ iff it is continuous at all points of the open interval $(a,b)$, from the right continuous at $a$ and from the left continuous at $b$.

Examples. The ceiling function $\lceil x\rceil $ is from the left continuous at each integer, the mantissa function $x-\lfloor x\rfloor $ is from the right continuous at each integer.

Title | one-sided continuity |

Canonical name | OnesidedContinuity |

Date of creation | 2013-03-22 17:57:50 |

Last modified on | 2013-03-22 17:57:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26A06 |

Related topic | OneSidedLimit |

Related topic | OneSidedDerivatives |

Related topic | OneSidedContinuityBySeries |

Defines | continuous from the left |

Defines | continuous from the right |

Defines | from the left continuous |

Defines | from the right continuous |

Defines | continuous on closed interval |