  Robustness measures are increasingly prominent resource quantifiers that have
been introduced for quantum resource theories such as entanglement and
coherence. Despite the generality of these measures, their usefulness is
hindered by the fact that some of their mathematical properties remain unclear,
especially when the set of resource-free states is non-convex. In this paper,
we investigate continuity properties of different robustness functions. We show
that their continuity depends on the shape of the set of free states. In
particular, we demonstrate that in many cases, star-convexity is sufficient for
Lipschitz-continuity of the robustness, and we provide specific examples of
sets leading to non-continuous measures. Finally, we illustrate the
applicability of our results by defining a robustness of teleportability and of
quantum discord.
