>>11434307>>11434339>>11434361What are you people on about?
The dirac delta is not a function that you can evaluate at points. It is often written as though it's a density for the lebesgue measure, like the point measure at 0, delta(x)dx, but this is shit notation created by engineers and physicists because they didn't understand what was going on.
The dirac delta functional is a linear functional on a given function space, usually a space of continuous functions over some pointed topological space. What it does is take a function and send it to the value of the function at the point. The linear functionals on such a space are in correspondance to a suitable set of measures on the space, and the delta's measure is the point measure at your point. i.e., sets have measure 1 if they contain the point and measure 0 otherwise. Integration with respect to this measure is evaluation of a function at the point.
Now, due to the radon nikodym theorem in L^1, you want to write this measure as h(x)dx for some standard measure on your space - for R this is obviously Lebesgue. But surprise surprise, you cannot do this with a point measure because point evaluation makes no sense in L^1.
But people do so anyway because they're idiots, and the whole theory of distributions (linear functionals on smooth function spaces) was developed as a band aid. Indeed, the delta is a distribution which acts on functions. It makes absolutely no sense to say delta(a) where a is a number, unless you mean the constant function at that number, in which case delta(a) = a and your identity doesn't hold. And if you meant for a, b, c to be functions, it is certainly not always true that a(0)b(0)c(0) = a(0) + b(0) + c(0).
