If you don't have a "handheld electrical tester" you can troubleshoot this particular problem with a piece of wire.

I'm assuming the gas tank is 1/2 full.

Attach one end of the wire to a good ground.

Turn on the key.

Touch the other end of the wire to the post on the gauge where the wire to the tank connects.

If the gauge still reads full the gauge is bad.

If the gauge reads empty, remove the test wire from the gauge and touch it to the post on the sending unit.

If the gauge still reads full the wire from the sending unit is bad.

If the gauge reads empty, either the sending unit is bad, or it has lost its connection to ground. Remove the test wire from the sending unit post and touch it to one of the screws holding the sending unit to the tank. If the gauge now reads the actual amount of gas in the tank (1/2 full) the ground for the tank is bad. If the gauge still reads full the sending unit is bad.

Turn off the key and remove the test wire.

'57 GMC 102, Original 347 V8, HydraMatic, 3.08 rear gear, added A/C, disk front brakes, HEI, AFB carb, '98 Honda Black Currant paint. T-boned and totaled 10/12
'52 GMC 152 Stake Bed, Original 228, SM420, added A/C, HEI, disk front brakes, '67 Chev 3.55 rear gear. Gets used as a real truck.