If $f$ is an H-lattice function (respectively a rational piecewise affine function) on $N_{\mathbb{R}}$, then there is a complete polyhedral complex $\Pi$ in $N_{\mathbb{R}}$ such that, for every $\Lambda\in\Pi$,

$$f|_{\Lambda}(u)=\langle m_{\Lambda},u\rangle+l_{\Lambda},$$

with $(m_{\Lambda},l_{\Lambda})\in M\times\mathbb{Z}$ (respectively $(m_{\Lambda},l_{\Lambda})\in M_{\mathbb{Q}}\times\mathbb{Q}$). Conversely, every piecewise affine function on $N_{\mathbb{R}}$ such that its defining affine functions have integral (respectively rational) coefficients, is an H-lattice function (respectively a rational piecewise affine function).