The Dirac delta function is zero everywhere except at the point where its argument is zero.
At that point, it is just the right kind of infinity so that
The transformation of an integral allows us to compute
If we make a wave packet in p-space using the delta function,
and we transform to position space,
This is a state of definite momentum written in momentum space.
Its Fourier transform is
This is a state of definite position written in position space.
Jim Branson 2013-04-22