I am a PhD student and during my research I was presented to the claim that

For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, one has $${\lim\inf}_{t\to\infty} f(x(t))=0\Longrightarrow {\lim\inf}_{t\to\infty} ||x(t)||=0.$$

In this context,

We say that a function $f:\mathbb{R}\to \mathbb{R}$ is positive definite if $f(x)\geq 0$ and $f(x)=0\iff x=0.$

We say that $p$ is a stable point at $t_0$ if, for any neighborhood $H$ of $p$, there's a neighboorhood $V$ of $p$ s.t. if $x(t_0)\in V$, then $x(t)\in H$ for all $t>t_0.$

I thought this is not true without more hypothesis. Could anyone have an ideia to (dis)prove this? If this is not true, any ideia about extra hypothesis? Thank you.