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We analyze Loewner traces driven by functions asymptotic to κ 1 − t. We prove a stability result when κ 6 = 4 and show that κ = 4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by κλ(t) are generated by a continuous curve for all κ> 0 with κ 6 = 4 but not when κ = 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 κ 1 − t and a conceptual proof of the corresponding results of Kager, Nienhuis an