Dini limits, singular primitives, and uniqueness

Dini integrability is sufficient, not necessary. For a fixed distribution $D$, $\delta \sin ({\delta }^{-2})D\to 0$ strongly, although its derivative contains $-2{\delta }^{-1}\cos ({\delta }^{-2})D$ with nonintegrable absolute logarithmic-scale pairing on a detecting test.