I am using the Chebyshev spectral differentiation technique that is described concisely under "details" here. The idea is to take the initial data $v_0,v_1\,...,v_N$ and store it in union with itself as the vector $$V = [v_0\,v_1\,...\,v_{N-1}\,v_N\,v_{N-1}\,v_{N-2}\,...\,v_1]^\top$$
From there, the Fourier transform of this vector $V$ is taken. However, for the Fourier transform to provide a good interpolation of the data in $V$, $V$ should be smooth and periodic. Although $V$ is continous and periodic, there is (generally) a discontinuity in its first derivative (around the entries $v_{N-1},v_N,v_{N-1}$). Why, then, is this method of differentiation still so effective?