The DFT is used when all you have available are samples of the function, rather than the function itself. If you are doing an FT on experimental data, it's always (as far as I know) recorded in discrete numbers: an array of floating point numbers, for example. There are a few times when the DFT has some applicability to real systems, for example simple theories of solids in which the ion cores occupy well-defined, regular, periodic locations. Not too many of those.
If you are using a computer to take a FT, and you know the mathematical form of the function, then you might be able to calculate the continuous FT symbolically using a computer algebra system. The output will be the mathematical form of the FT. If you calculate the FT "digitally", either because you have samples of experimental data, or you have evaluated a mathematical function a discrete regular points, then you are using the DFT and your output is an array of values.