Modify | Trim | Segment at Location

This trim tool breaks a line into segments at indicated points. The segments remain intact.

In the example below, the line is broken at the points indicated using the Segment at Location tool.

Indicating the break points for the segments

Duplicates

Duplicate objects hidden beneath line and circle

In the example below, the entire collection of curves was selected for repair.

Duplicates found and identified in selection

The line is divided into 3 segments at the points indicated above.

Line broken into 3 individual but connected segments

This tool is discussed and demonstrated in more detail here.