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

Tamed-adaptive Euler-Maruyama approximation for SDEs with superlinearly  growing and piecewise continuous drift, superlinearly growing and locally  H\"older continuous diffusion