Scribbling continua in Rn and constructing singularities with prescribed Nash fibre and tangent cone

Using approximations in the Hausdorff metric, we prove that any continuum in Rn is the set of limits at infinity of a real-analytic curve. This is then used to solve a problem concerning the characterization of the Nash fibre and the tangent cone of Whitney stratified sets at an isolated singularity: we show that every continuum (in the appropriate Grassmannian or sphere) can be realized as such.