O'Farrell, Anthony G.
(2004)
*When uniformly-continuous implies bounded.*
Bulletin of the Irish Mathematical Society, 53 (Summer).
pp. 53-56.
ISSN 0791-5578

## Abstract

Let (X,p) and (Y,σ) be metric spaces. A function f : X → Y is (by definition) bounded if the image of f has finite σ-diameter. It is well-known that if X is compact then each continuous f : X → Y is bounded. Special circumstances may conspire to force all continuous f : X → Y to be bounded, without Y being compact. For instance, if Y is bounded, then that is enough. It is also enough that X beconnected and that each connected component of Y be bounded. But if we ask that all continuous functions f : X → Y , for arbitrary Y, be bounded, then this requires that X be compact. What about uniformly-continuous maps? Which X have the property that each uniformly-continuous map from X into any other metric space must be bounded?

Item Type: | Article |
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Keywords: | Uniformly-continuous; Bounded; f : X → Y; Epsilon-step territories. |

Subjects: | Science & Engineering > Mathematics & Statistics |

Item ID: | 1812 |

Depositing User: | Prof. Anthony O'Farrell |

Date Deposited: | 26 Jan 2010 12:46 |

Journal or Publication Title: | Bulletin of the Irish Mathematical Society |

Publisher: | Irish Mathematical Society |

Refereed: | No |

URI: |

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