Consider a linear programming (LP) problem \begin{align} M(b) \in \arg\min_x \{ c^\top x : Ax=b, x \ge 0 \}. \end{align} Suppose the LP is feasible and bounded for all values of $b$. We know that $M(b)$ may not be a function, as $M(b)$ may not be unique. If at a given $b$, the LP has a unique solution, then "locally" M(b) is a linear function of $b$. This is because the basic feasible solution is $x_{B}=B^{-1}b$, where $B$ is the optimal basis. So, for sufficiently small changes in $b$, the optimal basis $B$ does not change, so the optimal solution will be $M(b+\hat{b})=B^{-1}b + B^{-1}\hat{b}$, where $\hat{b}$ is a small perturbation in $b$.
My question is what can be said for more global changes where the optimal basis changes? Does $M(b)$ have a piecewise linear behaviour?