the support of a distribution

The support of a distribution answers a simple question: where does it actually live? For an ordinary function, the support is the region where it is nonzero (more precisely, the closure of that region). For a distribution there are no pointwise values to look at, so we phrase the same idea through probes: a point lies outside the support if the distribution gives zero reading against every probe confined to a small neighbourhood of that point.

Precisely, a distribution T vanishes on an open set if (T, phi) = 0 for every test function phi supported inside that set. The support of T is what is left after you remove all such regions — the smallest closed set outside which T is zero. For the Dirac delta, every probe whose support avoids the origin gives reading zero, while a probe centred at the origin gives phi(0), which can be nonzero; so the support of delta is exactly the single point {0}. A distribution with support at one point turns out to be a finite combination of the delta and its derivatives there.

Support matters because it tells you the geometric footprint of a source or solution. A point source has support at one point; a charge spread over a surface has support on that surface; the fundamental solution of the wave operator in three dimensions has support exactly on a light cone, which is the rigorous statement of finite propagation speed and Huygens' principle. Knowing where a distribution is zero is often as useful as knowing what it is.