A function, eh?

Not a proper function, in the strict sense, but very useful all the same. It is defined by its properties when integrated.
    Definition: The Dirac delta function δ(x) satisfies (integral) δ(x) dx = 1 if the range of integration includes zero, =0 if not.
Take a moment to convince yourself that no R->R function satisfies this. Note that for any function continious at zero, (integral) f(x)δ(x) dx = f(0).

What use is it?

The neatest way to convince yourself of its necessity is to think of it as a continuum generalisation of the Kronecker delta symbol:
    (sum) fiδij = fj
    (integral) f(x)δ(x) dx = f(0)
As such, the notion of orthonormal eigenstates in Dirac's own formalism can be gerneralise from the discrete eigenvalue case to the continious case:
    <i|j> = δij becomes
    <x|y> = δ(x-y)
...which you need for such continious eigenvalues such as position and momentum. This, and the above stated property, make the sum-over-all-intermediate-states rule work out as it should:
    |x> = (integral) |y><y|x> dy

Generalisation to Rn:

    Definition: δ(n)(x) = δ(x1)δ(x2)...δ(xn)
...and then if you integrated this over some (n)volume, the result is 1 if zero is in the volume, 0 otherwise.

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