They are different... but normally it should be mainly a scaling differences...

Can refer to this blog post:

http://blogs.mathworks.com/steve/2010/01/18/relationship-between-continuous-time-and-discrete-time-fourier-transforms/

Hope that helps...

Yes, of course! FT is continuous in time and frequency, while DFT is discrete in time as well as in frequency. DFT is only an approximation.

If you want a good approximation, you'll have to use a high sampling rate for the discretization and you'll have to use a large number of samples, i.e. a large number of frequencies for which you calculate the DFT.

FFT is an effective Way of calculating the DFT, so the principle is the same.

Yes, these two transform return very different functions. They are not even similar. You may go to http://mizugadro.mydns.jp/t/index.php/FFT

and compare the blue line (which is Fourier Transform of the self-Fourier Gaussian) and the red segmented line (which is FFT of the same sample).

• A Discrete Fourier Transform is simply  the Fourier Transform when it is applied to discrete rather than a continuous signal. An Fast Fourier Transform is a faster version of the DFT that can be applied when the number of samples in the signal is a power of two. An FFT computation takes approximately N * log2(N) operations, whereas a DFT takes approximately N^2 operations, so the FFT is significantly faster.
• simply we can say FFT = Fast DFT

Yes, both are the separate transform methods.

The DFT (FFT just a fast implementation) operates on signals in the digital time domain.  The Fourier Transform (FT) operates on functions in the time continuous domain.  In order for there to be computability between the time digital and the time continuous time the time continuous has to be band limited (FT identically zero for frequencies above a given frequency Omega (Hz)) and the continuous wave form sampled at a rate at least 2*Omega samples per second.

In this case the Fourier Transform is realized as a periodic function and the DFT turns out to be a sampled version of the FT over a finite time set of samples.  However, the DFT is based on a finite number of samples while the FT is calculated using all samples.  Under certain conditions (ergodic or second order stationary stochastic process)  the DFT - if it is large enough will approximate the FT.  The example given earlier - Gaussian wave form - the above doesn't apply since the Gaussian waveform is not band limited. 

See, Kay, "Modern Spectral Estimation Theory and Applications," Prentice-Hall,  ISBN 0-13-598582-X

Well, addition to my previous note: The Fourier tansform (Fourier integral) can be approximated through the DFT. The special mispacement of sampel points is necessary. Every colleague, who thinks, that these are "almost the same" should consider at least one example, I suggest the self-Fourier function f(x)=exp(-x^2/2), and calculate the DFT of it, and compare to the Fourier transform of it.