Differentiate jumps without inventing point values

Try

A unit point source acts on a test function by reading its value at one location. At zero, δ(φ)=φ(0). This is a linear functional, not an ordinary function whose value is infinity at one point. Test functions here are smooth and compactly supported.

An ordinary locally integrable function f defines a distribution by φ↦∫fφ. Distributions extend this action-based viewpoint while requiring continuity in the standard test-function topology. That continuity condition is part of the definition, not an optional extra.

Predict

The distributional derivative is defined by T′(φ)=−T(φ′). Predict δ′(φ) when φ agrees with 5x+x² near zero. Explain why the answer depends on the derivative at zero rather than the value there.

Experiment

The answer is −5. For the Heaviside function H, equal to zero on the negative half-line and one on the positive half-line,

H′(φ)=−∫₀∞φ′(x)dx=φ(0).

Distribution	Action on φ
δ	φ(0)
δ′	−φ′(0)
δ″	φ″(0)
H′	φ(0)

Assigning any value to H at zero changes none of these integrals. Verify the alternating signs by applying the derivative definition twice rather than memorizing the table.

Understand

For a piecewise smooth function with jump J at a point, integration by parts on the two sides shows that its distributional derivative contains Jδ there, in addition to its ordinary derivative away from the jump. This explains the Rankine–Hugoniot calculation and the slope jump of a Green function using one common rule.

For E(x)=−|x|/2, the left derivative is 1/2 and the right derivative is −1/2. The jump is −1, so E″=−δ and −E″=δ. Thus E is a fundamental solution for −d²/dx². Adding any affine function preserves this identity, demonstrating that the source equation alone does not specify boundary behavior or growth conditions.

Delta cannot be represented by a locally integrable function. If a function represented it, testing on every region away from zero would force that function to vanish almost everywhere there. A single point has measure zero, so it would vanish almost everywhere altogether and could not evaluate a test to a nonzero value at zero. This uses the standard uniqueness property of distributions induced by locally integrable functions.

Not every formal operation extends to every distribution. Convolution is safely defined when one distribution has compact support; other cases need suitable support or decay conditions. Multiplication by a smooth function is defined, but multiplying two arbitrary distributions is not generally available. Tempered distributions act on Schwartz test functions and admit a Fourier transform by duality, extending the derivative and multiplier rules consistently.

Apply

Compute the distributional derivative of a piecewise affine function with one jump, identifying both the ordinary derivative and delta coefficient. Then find a continuous piecewise affine function whose second derivative is twice a delta. State whether your construction solves a free-space source identity or a boundary-value problem; those are separate specifications.

Continue

The free Schrödinger kernel is oscillatory rather than integrable in absolute value. Distributional and Fourier interpretations let us use it without pretending that it is a positive averaging density.