I intuitively understand what the DFT is doing and what the equation means. We are essentially adding up all the points of the input after projection onto the unit circle and then taking an average (dividing by 1/N). I note that this division of 1/N is often done in the IDFT (inverse).
What I don't understand is why when we consider the continuous fourier transform (integrated from -infinity to infinity), I notice there is no average. So in essence, this is just a sum.
Why do we need this 1/N factor in the DFT equation, but it omitted from both the forward and inverse continuous equations? Without it, isn't that just a sum? How mathematically is this equivalent to an average if we aren't dividing?
So it seems there is something weird about why scaling is necessary in DFT but not in the continuous case.
For example, equations 1 and 2 on the Wikipedia page about fourier transform (under definitions), shown the continuous transform, but there is no division by number of points. But isn't that just a sum?
Further, while I understand the 1/N in the DFT case, it isn't clear why we do it only on the inverse transform and why things are still correct in the forward transform without it.