= Distributional second derivative across a jump
{title2=$D^2y=y''_{\mathrm{reg}}+[y']_c\delta_c+[y]_c\delta'_c$}
Let $y$ be piecewise twice <continuously differentiable>, with finite one-sided values and derivatives at $c$, and write $[y]_c=y(c+)-y(c-)$. Its <distributional derivative> is $Dy=y'_{\mathrm{reg}}+[y]_c\delta_c$, and differentiating again gives
$$
D^2y=y''_{\mathrm{reg}}+[y']_c\delta_c+[y]_c\delta'_c.
$$
Here $\delta_c$ is the <Dirac delta> at $c$, and the subscript reg means the ordinary derivative on the two open intervals, interpreted as a <regular distribution>. To prove the first formula, split $-\int y\varphi'$ at $c$ and use <integration by parts> on each piece for a <test function> $\varphi$. The two boundary contributions add to $[y]_c\varphi(c)$. Applying the same argument to the regular first derivative proves the second formula. Thus a derivative jump produces a delta source, whereas a value jump produces the derivative of a delta source.
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