The reason why this is important is cause it tells you that if you want the Fourier series to converge pointwise, its limit has to be the average of its one-sided limits (granted, I have no clue what happens with functions where these don't exist, but that probably is actually unimportant). This explains part of the behavior of Fourier series at jump discontinuities. This result can also be adapted to give a rather quick proof of uniqueness of Fourier series (for sufficiently nice functions).
If you impose additional conditions on the one-sided derivatives of the function, you can improve c…