An affine transformation on $\mathbb{R}$ preserves mid-points. That is, if $f$ is affine then for any $x, z, q$, if $x$ is the mid-point of $[z, q]$, then $f(x)$ is the mid-point of $[f(z), f(q)]$.
Question: Are there any non-affine transformations on $\mathbb{R}$ with this same property? Or is it only affine transformations that preserve mid-points.