You are correct; the FFT or DFT is not the same thing as the continuous-time Fourier transform (CFT).
This technical report[1] examining the relation between the DFT and CFT shows in eq. 20 that the DFT is just the CFT evaluated at w=k2pi/(N*T). As N becomes large (number of pixels is large), this approaches the CFT.
I left this detail out; I wanted to put this in terms the reader knew.