Should I test a damaged extension cord or just replace it?

Replace it. A frayed cord can arc, start a fire, or give you a lethal shock. Continuity testing only checks if the copper wire is connected — it won’t reveal cracked insulation, loose plugs, or half-broken internal connections. Even if it passes a multimeter test, visual damage like cuts or wiggly prongs still makes it dangerous. A replacement costs ten bucks; don’t gamble your home or life.