  We study pathwise approximation of strong solutions of scalar stochastic
differential equations (SDEs) at a single time in the presence of
discontinuities of the drift coefficient. Recently, it has been shown by
M\"uller-Gronbach and Yaroslavtseva (2022) that for all $p \in [1, \infty)$ a
transformed Milstein-type scheme reaches an $L^p$-error rate of at least $3 /
4$ when the drift coefficient is a piecewise Lipschitz-continuous function with
a piecewise Lipschitz-continuous derivative and the diffusion coefficient is
constant. It has been proven by M\"uller-Gronbach and Yaroslavtseva (2020) that
this rate $3 / 4$ is optimal if one additionally assumes that the drift
coefficient is bounded, increasing and has a point of discontinuity. While
boundedness and monotonicity of the drift coefficient are crucial for the proof
of the matching lower bound of M\"uller-Gronbach and Yaroslavtseva (2020), we
show that both conditions can be dropped. For the proof we apply a
transformation technique which was so far only used to obtain upper bounds.

Probability