When you have a curve you can select one of the points and use the 'duplicate' function or you can select one of the middle points and use 'extrude curve and move'. Both of these things in 2.81 give me a solo unconnected point with two handles. What is this used for?
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1A curve is composed of one or more 'splines'. One spline cannot have branching out but can only be linear. That's why extruding can lead to a separated one. – lemon Oct 14 '19 at 05:38
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What Lemon said, see https://blender.stackexchange.com/questions/57284/it-is-possible-to-extrude-a-curve-to-two-directions/57288#57288 They are generally the start of anew curve but can have other uses like snapping target, or auxiliary geometry guide – Duarte Farrajota Ramos Oct 14 '19 at 16:47