For a linear program it is shown that the optimal policy set behaves continuously if the constraint vector changes on the set for which the program has a solution. The result implies that there exists a continuous optimal policy function for which a construction is indicated.
Bohm, V. (1975). On the continuity of the optimal policy set for linear programs. SIAM Journal on Applied Mathematics, 28(2), 303-306. https://hdl.handle.net/2078.5/149287 (Original work published 1975)