A damaged extension cord is not worth testing. Replace it.

Don’t. The risk isn’t subtle. A frayed cord can arc, start a fire, or give you a shock that stops your heart. Continuity testing will tell you if the copper wire is still connected, but it won’t tell you if the insulation is cracked, the plug is loose, or the internal connections are half-broken. That’s the kind of damage that fails when you least expect it.

If you really want to test for continuity, you need a multimeter set to resistance or continuity mode. Touch the probes to the two prongs on the plug — you should hear a beep or see near-zero resistance if the wire is intact. Same for the other end. But even if it passes, the cord is still dangerous if the jacket is cut or the prongs are wiggly. Visual inspection matters more than any electrical test.

Save the multimeter for other projects. A damaged cord costs ten bucks to replace. Don’t gamble your home or your life for that.