  Existence, uniqueness, and $L_p$-approximation results are presented for
scalar stochastic differential equations (SDEs) by considering the case where,
the drift coefficient has finitely many spatial discontinuities while both
coefficients can grow superlinearly (in the space variable). These
discontinuities are described by a piecewise local Lipschitz continuity and a
piecewise monotone-type condition while the diffusion coefficient is assumed to
be locally Lipschitz continuous and non-degenerate at the discontinuity points
of the drift coefficient. Moreover, the superlinear nature of the coefficients
is dictated by a suitable coercivity condition and a polynomial growth of the
(local) Lipschitz constants of the coefficients. Existence and uniqueness of
strong solutions of such SDEs are obtained. Furthermore, the classical
$L_p$-error rate $1/2$, for a suitable range of values of $p$, is recovered for
a tamed Euler scheme which is used for approximating these solutions. To the
best of the authors' knowledge, these are the first existence, uniqueness and
approximation results for this class of SDEs.
