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in my opinion, the limitation (or better referred to as condition that has to be fulfilled) for the differentation (derivative) property of the CTFT is that the CTFT has to be differentiable. This is like when you do derivative on functions in time domain, i.e. the function shall be differentiable in time domain, which usually needs the function only has anomaly (outlier, not well behaved) at only several places. For example unit pulse is no well behaved at the jumps, but it still could be differentiate.