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A new highly accurate fast algorithm is proposed for computing the Fourier transform integrals of discontinuous functions (DIFFT) by employing the analytical Fourier transforms of Gauss-Chebyshev-Lobatto interpolation polynomials and the scaled fast Fourier transform. This algorithm can achieve the exponential accuracy for evaluation of Fourier spectra at the whole frequency range with a low computational complexity. Furthermore, the algorithm allows the adaptive sampling densities for different sections of a piecewise smooth function. Numerical experiments are shown for the applications in computational electromagnetics.