For the first part, recall that for to be an isometry, we need to have that .
Now, if we choose and let , then we require that , that is, . Do you see how to continue?
"Let ( ) be a metric space where d(x,y) = |y-x|.
Let be an isometry from the metric space to itself.
Consider the image . Show that for every x in , the image equals or .
Next, show that assuming is continuous, either equals for every x, or it equals for every x."
First, I don't think I understand what "Consider the image " means. I guess it makes sense that each equals or , but I don't know how to prove it.
For the second statement, I know continuity means that for every there is a so when and that it's continuous for each x, but I don't know how to apply this definition to show that it is always either or for every x.
I really appreciate any help, even (or especially!) if it's just explaining very basic things that I probably don't understand correctly. Thanks.
Oh, now I understand that the question was just defining !
So from what you said, from , just solving for would lead to either the case of or by the nature of absolute values if I'm not mistaken.
Still, I don't know how to apply the definition of continuity though.
Thank you for your help!