Isometries of the reals We have been learning about isometries and how reflections, translations etc. and how they can affect a function. I was wondering if someone could help me with this proof by using the definition of isometries which I am struggling to understand. 

If $f: \mathbb R\to \mathbb R$ is an isometry of the reals. How do I show that $f$ is a reflection in a point if and only if $f$ has a unique fixed point. 

I must use definitions of isometries only, would that mean using other translations, reflections, and reflections? or how would I go about this.
I have been working through problems that practiced writing translations and reflections. For example, we have $g(x)=x+c$, $f(x)=2a-x$, and $h(x)=2b-x$, where $a$ and $b$ are the stationary points of $f$ and $h$ respectively, and $c$ is the amount we are translating by and the composition is 
$$
  h(g(f(x))) = h(g(2a-x)) = h(2a-x+c) = 2b-(2a-x+c) = 2(b-a+\frac{1}{2}c)-x
$$
so this is a reflection in the point $b-a+\frac{1}{2}c$.
But I am not quite sure this is the same work in this case.
 A: When tackling an "if and only if" proof, it will often seem that one direction is easier than the other. 
Here we are asked to assume that $f(x)$ is an isometry, then show it is a reflection if and only if it has exactly one fixed point.
Which is the easier direction?  It seems to me that proving that reflection of $\mathbb {R} $ in point $c $ has exactly one fixed point is very easy.
So let's focus on the more difficult implication, that if $f(x)$ is an isometry with one and only one fixed point, then it must be a reflection.
From already having the easy half done, we should expect that when $c $ is the unique fixed point, then the map $f:\mathbb {R}\to \mathbb {R} $ will turn out to be reflection in $c $.  How can we prove this.
It helps to write down what we want to show, and sometimes it is just about clear after we do.  Here the idea that $f $ is the reflection of real numbers in $c $ is expressed for all $x \in \mathbb {R} $:
$$ f(c+x) = c-x $$
Take a moment to think through why this identity expresses the notion of $f$ being a reflection in $c $.
Now try to argue, using the facts that $f $ is an isometry and that $c $ is the only fixed point, that this identity holds true for all real numbers $x $.
