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If the lamp uses a mains voltage bulb (of whatever sort), test continuity between the plug and the bulb holder with a multimeter on a low resistance range. To do this, touch one probe on one of the pins of the plug (but not the big earth pin) and see if you can get a zero reading by touching the other probe on one of the contacts in the bulb holder. The switch may be off. Try it in the other position of the switch. Now repeat with the probe on the other pin of the plug. If you can't get a zero reading in either case then there is a break in the wire, or the switch is faulty.