jump condition
Picture pinching a taut string at one point and tugging it sideways. The string stays continuous — it does not tear — but it develops a sharp corner at the pinch: the slope on the left differs from the slope on the right. That sudden kink is the visible fingerprint of a point force, and the jump condition is the rule that fixes exactly how big the kink must be.
It is the device that builds a Green's function for an ODE. To solve L G = delta(x - s) you note that away from the source point s the right-hand side is zero, so G solves the homogeneous equation on each side — choose a homogeneous solution satisfying the left boundary for x < s and one satisfying the right boundary for x > s. Two conditions then splice them at x = s. First, G is continuous there. Second, integrating L G = delta across an infinitesimal interval around s shows the highest derivative must jump: for L = -d/dx(p dG/dx) + ..., the slope satisfies G'(s+) - G'(s-) = -1/p(s). Continuity plus this prescribed derivative jump pin down the two unknown constants and complete G.
The jump condition is where the Dirac delta does its work concretely. The delta is invisible except through the discontinuity it forces in a derivative; the size of that discontinuity is dictated by the coefficient of the highest derivative in L. The same accounting reappears across physics: the kink in the electrostatic potential's slope at a surface charge, the jump in the wavefunction's derivative at a delta-potential in quantum mechanics, the corner in a beam's slope under a point load.
For -G'' = delta(x - s) on [0, 1] with G(0) = G(1) = 0: take G = A x for x < s and G = B(1 - x) for x > s. Continuity at s gives A s = B(1 - s); the jump G'(s+) - G'(s-) = -1 gives -B - A = -1. Solving, A = 1 - s, B = s, recovering G(x, s) = x(1 - s) for x < s.
Continuity plus a unit jump in the first derivative pins down both constants — the standard recipe for an ODE Green's function.
Which derivative jumps depends on the order of the operator: for a second-order L the function stays continuous and only its first derivative jumps. For a fourth-order beam operator the function and its first two derivatives stay continuous and the third derivative jumps — read the order off L before applying the condition.