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.
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 $.