Hi guy's I know this is more of a homework question, I posted a similar thread earlier on but I think I ended up confusing myself.(adsbygoogle = window.adsbygoogle || []).push({});

I need to show that a function is continous between metric spaces. I'll post the question and what i've done any tips on moving forward would be great.

I have any metric spaces

[itex] (X,\rho) [/itex]

and

[itex](Y, \theta)[/itex]

And a metric space

[itex](X,\bar\rho)[/itex]

where

[itex]

\bar\rho:X \times X \Rightarrow R_{0}^{+}, (x,y) \Rightarrow \frac{\rho(x,y)}{1+\rho(x,y)}.

[/itex]

I have got to show the following

Let [itex](Y, \theta)[/itex]

be a metric space.

Prove that.

[itex] f : X \rightarrow Y [/itex]

is continuous with respect to [itex]\bar\rho[/itex] if and only if it is continuous with respect to [itex]\rho[/itex]

I have been given that [itex] f : X \rightarrow Y [/itex]

is continuous with respect to [itex](X,\rho)[/itex]

So I know that for some [itex]\delta[/itex] and [itex] \epsilon > 0[/itex]

that

[itex]{\rho}(z,b) < \delta \rightarrow \theta(f(z),f(b)) < \epsilon[/itex]

I need to show that for some[itex] \psi > 0[/itex] that

[itex]{\bar\rho}(x,a)<\psi \rightarrow \theta(f(x),f(a)) <\epsilon[/itex]

Can some one please show me how to go about finding [itex]\psi[/itex] ?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuous functions in metric spaces

Loading...

Similar Threads - Continuous functions metric | Date |
---|---|

Looking for a Theorem of Continuous Functions | Nov 8, 2012 |

Convergent Filter Base and Continuous Function | Aug 12, 2009 |

Hello! How to prove the min function is continuous? | Oct 27, 2008 |

The space of continuous functions. | Feb 19, 2008 |

Continuous function | Aug 7, 2007 |

**Physics Forums - The Fusion of Science and Community**