  Recently a lot of effort has been invested to analyze the $L_p$-error of the
Euler-Maruyama scheme in the case of stochastic differential equations (SDEs)
with a drift coefficient that may have discontinuities in space. For scalar
SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion
coefficient that is non-zero at the discontinuity points of the drift
coefficient so far only an $L_p$-error rate of at least $1/(2p)-$ has been
proven. In the present paper we show that under the latter conditions on the
coefficients of the SDE the Euler-Maruyama scheme in fact achieves an
$L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$ as in the case of
SDEs with Lipschitz coefficients.
