constant prefactors and scaling in continuous and discrete fourier transforms

Question

Depending on the definitions, the constant prefactors in continuous and discrete fourier transforms can differ arbitrarily as long as the forward and backward counterparts multiply up to 1/T (period) or 1/2$\pi$ (fourier series/fourier transform). But most textbooks seem inconsistent in defining them for continuous and discrete fourier transforms, in that for instance for continuous 1/2$\pi$ is with the frequency amplitude and for discrete 1/N (# of sampling points) is with the temporal amplitude. Moreover, dimensionality or scaling balance is broken in all definitions as far as I know. How to understand all these and is there a good text with consistency?