  The aim of this paper is to study weak and strong convergence of the
Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic
differential equation $\mathrm{d} X_t=\sigma(X_t) \mathrm{d} W_t$ with
non-sticky condition. For proving this, we first prove that the Euler--Maruyama
scheme also satisfies non-sticky condition. As an example, we consider
stochastic differential equation $\mathrm{d} X_t=|X_t|^{\alpha} \mathrm{d}
W_t$, $\alpha \in (0,1/2)$ with non-sticky boundary condition and we give some
remarks on CEV models in mathematical finance.
