Proposition: Flat multipliers can fail to repair extension

Define

Then $v$ is smooth and flat at zero, but

is nonextendable.

Proof. Every derivative of $e^{-2/x^2}$ is a finite sum of powers of $x^{-1}$ multiplied by $e^{-2/x^2}$, and tends to zero at zero. The same estimates after division by any fixed power of $x$ give the smooth flat extension. The product identity holds on the punctured interval, and Proposition prop:nonextendable applies.