Definition 8.5.7.9 (Dyadic Homeomorphisms). Recall that a dyadic rational number is a real number of the form $\frac{a}{2^ n}$, where $a$ and $n$ are integers. Let $s, t \geq 0$ be dyadic rational numbers. We say that a homeomorphism $f: [0,s] \xrightarrow {\sim } [0,t]$ is dyadic if it satisfies the following conditions:
The function $f$ is piecewise linear; in particular, it is differentiable away from finitely many points of the closed interval $[0,s]$.
If $x \in [0,s]$ is a point where $f$ is not differentiable, then $x$ is a dyadic rational number.
For every point $x \in [0,s]$ where $f$ is differentiable, the derivative $f'(x)$ is equal to $2^{n}$ for some integer $n$.
Note that the third condition implies that the homeomorphism $f$ is strictly increasing, so that $f(0) = 0$ and $f(s) = t$.