  In this paper, we consider stochastic differential equations whose drift
coefficient is superlinearly growing and piece-wise continuous, and whose
diffusion coefficient is superlinearly growing and locally H\"older continuous.
We first prove the existence and uniqueness of the solution to such stochastic
differential equations and then propose a tamed-adaptive Euler-Maruyama
approximation scheme. We study the rate of convergence in the $L^1$-norm of the
scheme in both finite and infinite time intervals.
