Atoms obstruct a continuous distributional limit

If $Z_n$ has probability at least p at $z_n$, with $p>0$ and $z_n\to z$, every weak limit has probability at least p at z. Apply the closed-set bound of the Portmanteau theorem to arbitrarily small closed intervals containing z and eventually $z_n$, then decrease their radii. Thus a persistent atom prevents convergence to a continuous law, including a chi-squared law. This provides a direct check when bounded-information estimation fails standard asymptotic-normal theory.