Let $\mathscr{X}$ be a strictly semistable model of $X$ and let $f\colon{X^{{\mathrm{an}}}}\to\mathbb{R}$ be a function. Then the following properties hold:

• (a)

  If $f$ is a $\mathbb{Z}$-model function, then $f|_{S(\mathscr{X})}$ is a piecewise linear function which is integral $\Gamma$-affine.

• (b)

  The function $f$ is a $\mathbb{Z}$-model function determined on $\mathscr{X}$ if and only if $f=F\circ\tau$ for some function $F\colon S(\mathscr{X})\to\mathbb{R}$ which is affine on each edge of $S(\mathscr{X})$ with integer slopes and with $f(v)\in\Gamma$ for each vertex $v$ of $S(\mathscr{X})$.

• (c)

  If $G$ is a piecewise linear function on $S(\mathscr{X})$ which is integral $\Gamma$-affine, then $G\circ\tau$ is a $\mathbb{Z}$-model function.