Especially in math, a distinction is often drawn between FT f(real) -> F(real) DFS f(real) -> F(int) DTFT f(int) -> F(real) DFT f(int) ->

Especially in math, a distinction is often drawn between FT f(real) -> F(real) DFS f(real) -> F(int) DTFT f(int) -> F(real) DFT f(int) -> F(int) Only one of these, the DFT, is easily handled by a computer, but they all have an important role in both theory and practice. For example, why does JPEG use a Discrete Cosine Transform rather than a Discrete Fourier Transform? What are the pitfalls of approximating a continuous domain with discrete samples? Why are engineers obsessed with window functions? A continuous domain will lead you to the right answer. You can't easily understand the DFT (or FFT) without the FT and DFT. DFS has its own uses. To many (applied) mathematicians, the fact that a computer uses the FFT algorithm to compute a DFT is largely immaterial and not particularly worth mentioning, because they are more interested in the questions in the preceding paragraph than questions like "how many points can I take on X hardware while maintaining Y performance?"