Differentiable comparative statics with payoff functions not differentiable everywhere

In this note we consider Cournot oligopoly. Due to the consideration of several consumers and/or technologies, the profit function is not differentiable everywhere. We show that Cournot equilibrium never occurs in outputs where profit functions are not-differentiable. This result validates the procedure of differentiating profit functions to perform comparative statics in models where profit functions are not differentiable everywhere.