You have a function defined on the boundary of $\mathbb R^n_{\ge 0}$. Its restriction to each face is smooth. You can extend it to all of $\mathbb R^n_{\ge 0}$ by writing $$ f(x)=\sum_P (-1)^{|P|-1}f_P(x_P), $$ where the $x\mapsto x_P$ is the projection $\mathbb R^n_{\ge 0}\to C_P$.

You have a function defined on the boundary of $\mathbb R^n_{\ge 0}$. Its restriction to each face is smooth. You can extend it to all of $\mathbb R^n_{\ge 0}$ by writing $$ f(x)=\sum_P (-1)^{|P|-1}f_P(x_P), $$ where $x\mapsto x_P$ is the projection $\mathbb R^n_{\ge 0}\to C_P$.