Let's assume that you want to join 2 lines in QGIS:
Activate edition mode and click on the magnet tool:
Click on the first vertex and bring the cursor close to the second one:
At this point, you'll have a continuous line. Finally, dissolve attributes of both lines to have only one entry on the attribute table. Go to:
Vector > Geoprocessing tools > Dissolve
Here is the final result:
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