<p>In this paper, we investigate, from a variational analysis standpoint, the so-called <i>Set of Sustainable Thresholds</i> associated with a given control system. This set corresponds to a certain collection of parameters that ensure, for a given initial state, that the pathwise constraints of the control systems are satisfied within a prescribed period of time. Our goal is to study properties of the Set of Sustainable Thresholds when it is seen as a set-valued map that depends on the initial state of the control system. The novelty of this work is that our analysis is carried out in the context of continuous-time control systems, extending recent developments reported for the discrete-time framework. Using tools from differential inclusions, we establish conditions for ensuring that the Set of Sustainable Thresholds defines a closed set-valued map. Furthermore, we investigate convexity properties of this set-valued map by using some recent developments in monotonicity analysis for nonlinear dynamical systems. Finally, we study continuity properties of this set-valued mapping, with emphasis on lower semicontinuity and Lipchitz continuity.</p>