Regularity of $f\,dt$ at a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$: exactly when $f$ is in the local ring

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Regularity of $f\,dt$ at a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$: exactly when $f$ is in the local ring · Prove2Me