Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be continuous and suppose $0$ is a fixed point of $f$ (that is, $f(0) = 0$). Does $W^u(0) = \varnothing$ imply that $0$ is a Lyapunov stable fixed point of $f$? My guess is yes, but how can it be proved? If not, what is a counterexample? This seems rather fundamental, but I cannot find a solution online.
Definitions: the unstable manifold of $0$, denoted $W^u(0)$, is the set of all $x \ne 0$ for which there exists a backwards orbit with $f^i(x) \to 0$ as $i \to -\infty$. The fixed point $0$ is said to be Lyapunov stable if for every neighbourhood $U$ of $0$ there exists a (smaller) neighbourhood $V$ of $0$ such that $f^i(x) \in U$ for every $x \in V$ and every $i \ge 0$.