FTP, name these generalized functions equal to the Dirac delta function under the operation of an associated operator and derived from their namesake's second identity.
In one-space, these are generally continuous but have kinks where the source and field points coincide.  These distributions may be used to create Friedholm integral equations from PDE's since their spatial convolution with the density function is equal to the unknown function, and those associated with self-adjoint operators have the useful property of symmetry, so that the source and field vectors may be interchanged.  FTP, name these generalized functions equal to the Dirac delta function under the operation of an associated operator and derived from their namesake's second identity.
In one-space, these are generally continuous but have kinks where the source and field points coincide.  
Green's functions