Don't guess with extension cords.
Short answer: you can’t reliably test for internal breaks without a continuity tester — which is basically what a multimeter does. But if you absolutely need to check right now, there’s a workaround that finds obvious breaks without one.
Plug in a lamp or small appliance that you know works. Then, while it’s running, slowly bend the cord along its entire length, paying close attention near the plug ends and any spots where it’s been kinked or walked over. If the light flickers or cuts out when you bend a certain spot, you found the break. That’s a “flex test.”
This is finicky and will miss partial breaks (high resistance without an open circuit). It also won’t tell you if the cord is dangerously hot but still passing current. And obviously, be careful — you’re messing with a live cord. Don’t do this near water or with kids around.
The flex test is a quick diagnostic, not a safety pass. If the cord is old, frayed, smells funny, or feels warm during normal use, just replace it. They’re cheap. Future you doesn’t want to find the break the hard way by smelling smoke.