"Let ($\displaystyle \mathbb{R}, d$) be a metric space where d(x,y) = |y-x|.

Let $\displaystyle \phi:\mathbb{R}\rightarrow\mathbb{R}$ be an isometry from the metric space to itself.

Consider the image $\displaystyle x_0 = \phi(0)$. Show that for every x in $\displaystyle \mathbb{R}$, the image $\displaystyle \phi(x)$ equals $\displaystyle x_0 + x$ or $\displaystyle x_0 - x$.

Next, show that assuming $\displaystyle \phi$ is continuous, either $\displaystyle \phi(x)$ equals $\displaystyle x_0 + x$ for every x, or it equals $\displaystyle x_0 - x$ for every x."

First, I don't think I understand what "Consider the image $\displaystyle x_0 = \phi(0)$" means. I guess it makes sense that each $\displaystyle \phi(x)$ equals $\displaystyle x_0 + x$ or $\displaystyle x_0 - x$, but I don't know how to prove it.

For the second statement, I know continuity means that for every $\displaystyle \epsilon > 0, $ there is a $\displaystyle \delta > 0, $ so $\displaystyle d(\phi(x), \phi(x_0)) < \epsilon $ when $\displaystyle d(x, x_0) < \delta $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 $\displaystyle x_0 + x$ or $\displaystyle x_0 - x$ 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.