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We define a differential game of dynamic public investment with a discontinuous Markovian strategy space. The best response correspondence for the game is well-behaved: best responses exist and uniquely map almost all profiles of opponents’ strategies back to the strategy space. Our chosen strategy space thus makes the differential game well-formed, resolving a long-standing open problem and allowing the analysis of a wider class of differential games and Markov-perfect equilibria. We provide a ‘cookbook’ necessary and sufficient condition for constructing the best response, and demonstrate its use with a canonical model of non-cooperative mitigation of climate change. Our approach provides novel, economically important results: we obtain the entire set of symmetric Markov-perfect Nash equilibria, and demonstrate that the best equilibria can yield a substantial welfare improvement over the equilibrium which previous literature has focused on. Our methods do not require specific functional forms.