Let $\Psi$ be a virtual support function on $\Sigma$ and let $\psi$ be a rational piecewise affine function on $N_{\mathbb{R}}$, with $\operatorname{rec}(\psi)=\Psi$, such that there exists a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ piecewise affine on $\Pi$. Then $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$.