Proposition: Smoothness off a locus does not ensure extension

A smooth distribution on a punctured domain need not extend across the puncture.

Proof. Take $e^{1/x^2}$ on $0<x<1$. A distribution extension near zero has finite order on every fixed compact interval. Evaluate it on a nonnegative bump centered at $\epsilon$ with width proportional to ${\epsilon }^2$. Its test seminorms grow polynomially in ${\epsilon }^{-1}$, but its pairing is bounded below by $c{\epsilon }^2{e}^{c^{\prime }/{\epsilon }^2}$. This contradicts finite-order continuity.