# continuous

Let $X$ and $Y$ be topological spaces^{}. A function $f:X\to Y$ is continuous^{} if, for every open set $U\subset Y$, the inverse image ${f}^{-1}(U)$ is an open subset of $X$.

In the case where $X$ and $Y$ are metric spaces (e.g. Euclidean space^{}, or the space of real numbers), a function $f:X\to Y$ is continuous if and only if for every $x\in X$ and every real number $\u03f5>0$, there exists a real number $\delta >0$ such that whenever a point $z\in X$ has distance less than $\delta $ to $x$, the point $f(z)\in Y$ has distance less than $\u03f5$ to $f(x)$.

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space $X$ at once). When $X$ and $Y$ are topological spaces, we say $f$ is continuous at a point $x\in X$ if, for every open subset $V\subset Y$ containing $f(x)$, there is an open subset $U\subset X$ containing $x$ whose image $f(U)$ is contained in $V$. Of course, the function $f:X\to Y$ is continuous in the first sense if and only if $f$ is continuous at every point $x\in X$ in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).

In the common case where $X$ and $Y$ are metric spaces (e.g., Euclidean spaces), a function $f$ is continuous at $x\in X$ if and only if for every real number $\u03f5>0$, there exists a real number $\delta >0$ satisfying the property that $$ for all $z\in X$ with $$. Alternatively, the function $f$ is continuous at $a\in X$ if and only if the limit of $f(x)$ as $x\to a$ satisfies the equation

$$\underset{x\to a}{lim}f(x)=f(a).$$ |

Title | continuous |

Canonical name | Continuous |

Date of creation | 2013-03-22 11:51:55 |

Last modified on | 2013-03-22 11:51:55 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 12 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 26A15 |

Classification | msc 54C05 |

Classification | msc 81-00 |

Classification | msc 82-00 |

Classification | msc 83-00 |

Classification | msc 46L05 |

Synonym | continuous function |

Synonym | continuous map |

Synonym | continuous mapping |

Related topic | Limit |

Defines | continuous at |