# Thread: Showing f is continuous

1. ## Showing f is continuous

Hey,

I am able to show how f is continuous if the domain is a discrete space. However, in this case the domain is a metric space. Therefore what would be the easiest way to show that f is continuous in this case.

Thanks

http://i53.tinypic.com/iemnv6.jpg

2. ## Re: Showing f is continuous

Show that $\displaystyle f$ is Lipschitz continuous thanks to the triangular inequality.

3. ## Re: Showing f is continuous

Originally Posted by mathshelpee
Hey,

I am able to show how f is continuous if the domain is a discrete space. However, in this case the domain is a metric space. Therefore what would be the easiest way to show that f is continuous in this case.Thanks
http://i53.tinypic.com/iemnv6.jpg
Do you know the theorem: If $\displaystyle a,~b,~\&~x_0$ are points in a metric space $\displaystyle X$ then $\displaystyle \left| {d(a,x_0 ) - d(b,x_0 )} \right| \leqslant d(a,b)$.

Using that you can see if $\displaystyle \varepsilon > 0$ let $\displaystyle \delta=\varepsilon/2$.
Now if $\displaystyle d(a,b)<\delta$ then $\displaystyle \left| {d(a,x_0 ) - d(b,x_0 )} \right| \leqslant d(a,b)< \varepsilon .$