Stability of the solution to stochastic delay differential equations (SDDEs) have received a great deal of attention, but there is so far little work on stability of numerical solutions to nonlinear stochastic functional differential equations (SFDEs). To close the gap, this paper proposes and studies a numerical scheme called complete backward Euler numerical scheme for general SFDEs. In the paper, we come up with a more general polynomial growth condition with Lyapunov function. Under the generalized polynomial growth condition, the almost sure exponential stability of the underlying continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Besides, the solvability of the implicit scheme is studied specially by introducing the concept of the generalized monotone vectorial functions. To establish the stability criteria for the nonlinear continuous model and the implicit scheme, some necessary lemmas have been established at first. At the end of the paper, a numerical example with simulation is proposed to illustrate the conditions, method and conclusions of the paper.

随机时滞微分方程（SDDE）的解的稳定性受到了广泛的关注，但到目前为止，关于非线性随机泛函微分方程（SFDE）的数值解的稳定性的工作还很少。为了弥补这一差距，本文提出并研究了一种用于通用SFDE的数值方案，称为完全后向Euler数值方案。在本文中，我们提出了一个具有Lyapunov函数的更一般的多项式增长条件。对比研究了广义多项式增长条件下底层连续模型和数值格式的几乎确定的指数稳定性。可以确认，该数值方案保持了连续模型的稳定性，而对步长没有限制。除了，通过引入广义单调矢量函数的概念，专门研究了隐式方案的可解性。为了建立非线性连续模型和隐式方案的稳定性准则，首先建立了一些必要的引理。最后，给出了一个带有仿真的数值例子来说明本文的条件，方法和结论。