Suppose we have a $(X,d)$ a metric space which is not bounded. Then take $f,g : \Bbb R \rightarrow \Bbb R$ both defined with $f(x)=g(x)=x$. Then $f$ and $g$ are Lipschitz, but $(fg)(x)=x^2$ is not Lipschitz.
Could someone please give me more such examples where there are two Lipschitz functions but their product is not Lipschitz given $(X,d)$ is not bounded.