Buy a multimeter. It's the only reliable way.

Use a multimeter set to continuity or low resistance (ohms). Plug nothing in. Test each conductor from one end to the other — live to live, neutral to neutral, ground to ground. If you get infinite resistance or no beep, the wire is broken inside. If you get any reading above a few ohms for a short cord (say, >1 ohm for a 6-footer), there’s corrosion or a partial break, and the cord is unsafe under load.

You can also try the old lamp trick: plug in a small light and wiggle the cord along its length. If the light flickers, you’ve found the break. But that doesn’t tell you about high resistance — it just tells you it’s broken enough to notice. A multimeter gives you numbers, and numbers don’t lie.

If you don’t have a multimeter or don’t want to learn how to use one, just replace the cord. A new extension cord costs less than twenty bucks. Testing internal breaks is a nice skill to have, but not worth burning your house down over a bad splice.