All point don't necessary have to hane control handles and the direct selection tool can select a segment in a path or sub-path and may only show the control handles at both ends of the selected segment. Here is a path the has four closed elliptical sub-paths each sub-path have four control points with handles and four segments. Here I use the direct selection tool and clicked on a segment in one of the four closed sub-path. The four control points in the closed sub-path were then shown an the control handle on both ends of the clicked segment are also show.
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All select with no control points showing.
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Selecting many segments showing many handles
