Asymptotic stability of the zero solution of \[\dot x(t) = - a(t)x(t) + P(t,x_t )\] is studied with a direct method of Razumikhin. If $| {P(t,\varphi )} | \leqq \alpha (t)\theta \| \varphi \|$ for some $\theta 0$, then zero is exponentially stable; if the condition on $a(t)$ is weakened to $a(t) \geqq 0$ and $\int ^\infty a(t)dt = \infty $, then zero is asymptotically stable.