Background to my question: I'm taking a signals and systems course where we are using the Dirac delta function, but since it's an engineering course the explanation of what it actually is has been very hand-wavy with just enough for us to be able to do transforms with it.
It was first explained to me that the Dirac delta function is an impulse of infinite height and infinitely small width, so that the area under the impulse ends up being equal to 1. Then we started solving for the Fourier transforms of functions and the area under these impulses has taken many different values. Now if the area under the impulse is no longer 1, is the width no longer infinitely narrow so that it extends past its specific point on the frequency axis? Or is the height greater than infinity? Or something else?