Distributional second derivative across a jump ($D^{2}y=y_{\mathrm{reg}}^{\prime \prime }+[y^{\prime }{]}_c{\delta }_c+[y{]}_c{\delta }_c^{\prime }$)

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 $Dy=y_{\mathrm{reg}}^{\prime }+[y{]}_c{\delta }_c$, and differentiating again givesHere ${\delta }_c$ 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 $-\int y{\phi }^{\prime }$ at $c$ and use integration by parts on each piece for a test function $\phi$. The two boundary contributions add to $[y{]}_c\phi (c)$. 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.