The class of continuous nonlinear controllers that are based on control Lyapunov functions (clfs) can be understood graphically as continuous selections from a set-valued map that takes states and maps them to control-value sets. The notion of inverse-optimality and robustness to input disturbances can also be graphically understood by this set-valued map. The graphical approach introduced in this paper makes clear the relationship between Sontag's formula, Freeman and Kokotovic' min-norm formula, and other 'universal formulas', as well as shedding light on the meaning of Lvapunov redesign. Most importantly, this paper examines a null-space associated with these set-valued maps which offers the potential for significant improvement in the large-signal performance of Lyapunov-based controllers.