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I need to show that a function is continuous 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] ?