= Jump contribution to a piecewise weak derivative
{title2=$Du=u'_{\rm pieces}+[u]_a\delta_a$}
For a <function> smooth on either side of a joining point $a$, <integration by parts> on the two <intervals> adds a Dirac mass whose coefficient is the right <Sobolev trace>[trace] minus the left <Sobolev trace>[trace]. If the one-sided <derivatives> are <locally integrable>, a first <weak derivative> can be represented by a <locally integrable function> across the join only when the jump vanishes. <Derivative> jumps themselves do not obstruct first-order Sobolev membership: a <continuous> piecewise <Lipschitz function> still has a <bounded> weak first <derivative>.
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