We analyze Loewner traces driven by functions asymptotic to $\kappa \sqrt{1-t}$. We prove a stability result when $\kappa \ne 4$, 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 $\kappa \ne 4$, 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.