Rectangular vs Hann windowing
When you filter a finite length signal, the filtered output will have a larger length, either you implement the filter in time domain or frequency domain. Your desired continuity comes from this excess part which must be added to the result of next segment or window.

For example when you take N samples of a signal and filter it with a filter of length M, the result has a length of N+M-1, where the M-1 excessive samples must be added to the next segment. To do so you have to find your filter response in time domain which has a length of M samples, then zero-pad both your signal and filter to N+M-1 samples then take N+M-1 point FT of both and multiply them.