Let's assume that you want to know if a line is continuous in QGIS. If a line has no discontinuities, the last vertex of every segment should intersect with the first vertex of the next element.
Click on the vertex editor icon:
Do a right-click on the line to open the vertex editor table (select the first row if you wish):
In this example, there is a green X mark in the middle of the line. Therefore, the polyline isn't continuous.
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