A local Euler convention

For a first-order density this becomes

For a second-order density, first and second derivatives of the ramp may occur.

With the fixed ${\sigma }_{\epsilon }$, these layer terms vanish outside the closed layer $0\le t\le \epsilon$ and are smooth and flat at its endpoints. Their nonzero values lie in $0<t<\epsilon$. They are not sharp-surface distributions.