  We show that the knot type of the link of a real analytic map germ with
isolated singularity $f\colon(\mathbb{R}^2,0)\to(\mathbb{R}^4,0)$ is a complete
invariant for $C^0$-$\mathscr A$-equivalence. Moreover, we also prove that
isolated instability implies $C^0$-finite determinacy, giving an explicit
estimate for its degree. For the general case of real analytic map germs,
$f\colon (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^p,0)$ ($n \leq p$), we use
the Lojasiewicz exponent associated to the Mond's double point ideal $I^2(f)$
to obtain some criteria of Lipschitz and analytic regularity.
