Jump contribution to a piecewise weak derivative ($Du=u_{\mathrm{pieces}}^{\prime }+[u{]}_a{\delta }_a$)

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 trace minus the left 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.