Let $D\subset\mathbb{R}^n$ be open, let $f\in C^1(D;\mathbb{R}^n)$, and let $x^*\in D$ be an equilibrium such that every eigenvalue of $Jf_{x^*}$ has negative real part. Then there exist open neighbourhoods $W\subsetneq U\subsetneq D$ of $x^*$ with $\overline{U}$ compact and $\overline{U}\subset D$, and a constant $\varepsilon>0$ such that the following holds: if $g\in C^1(U;\mathbb{R}^n)$ satisfies

\begin{align*} \sup_{x\in U}|g(x)-f(x)|+\sup_{x\in U}\|Jg_x-Jf_x\|<\varepsilon, \end{align*}

then there exists a unique point $x_g\in W$ with $g(x_g)=0$, where uniqueness is among equilibria lying in $W$. If $(\psi_t)$ denotes the maximal forward local flow generated by $\dot{x}=g(x)$ on $U$, then there is an open neighbourhood $W_g\subset W$ of $x_g$ such that every $x_0\in W_g$ satisfies $\psi_t(x_0)\in U$ for every $t\ge 0$, the trajectory is defined for every $t\ge 0$, and $x_g$ is an asymptotically stable equilibrium with respect to this restricted forward flow.