Suppose $f: \Bbb R \rightarrow \Bbb R$ is an isometry of the reals.
Prove that $f$ is a symmetry about a point if and only if $f$ has a unique fixed point.
Part 1: The assumption is $f$ is a symmetry and we want to conclude it has a unique fixed point.
Part 2: The assumption is $f$ is an isometry with a unique fixed point and we want to conclude $f$ is a symmetry.
Definition: A fixed point of $f$ is $x$ such that $f(x)=x$.
I'm not sure the best way to approach this problem, I think for both parts a proof by contradiction would be easiest, however I am not sure how do either of these.
For part $1$, use the fact that a symmetry is a mirroring of $\mathbb R$ in a point, so that point must be fixed.
For part $2$, show that the isometry is not a translation. Then, deduce that is must be a symmetry.