We analyze Loewner traces driven by functions asymptotic to $\kappa\sqrt{1-t}$ . We prove a stability result when $\kappa\neq4$ , and we show that $\kappa=4$ can lead to nonlocally connected hulls. As a consequence, we obtain a driving term $\lambda(t)$ so that the hulls driven by $\kappa \lambda(t)$ are generated by a continuous curve for all $\kappa>0$ with $\kappa\neq 4$ , but not when $\kappa=4$ , so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by $\kappa\sqrt{1-t}$ and a conceptual proof of the corresponding results of Kager, Nienhuis, and Kadanoff.