Broken Sword can be repaired to its original weapon at the blacksmith. Check the original weapon for the repair cost.

The Broken Sword is what all Swords become when it loses all durability. You should try to repair or forge your Broken Swords, but there is utility with Broken weapons. Because of their low Mt and Hit, they are exceptionally good at farming Sword ranks and Class Mastery (though unequipping is better for just Class Mastery) in Auxiliary battles, as your units will barely deal any damage to enemy units.