From the looks of the pictures I would say that it needs a rewiring it looks like some of the insulation is chared.
A continuity test can be picked up at almost any hardware store, usually less than $10. It generally looks like a pointed screwdriver with a wire coming out of the handle with a clip on the end of the wire. Inside the handle is a light and a battery, if you touch the clip on the wire to the pointed blade the light is on which means the battery good . You would attach the clip on one of the two wires of the lamp, you would then touch the outer threaded part of the light socket, if the tester lights the wire should be considered the common or white wire. If the tester did not light touch the tester to the center button down inside the light socket. If the tester then lights that means you are connected to the hot or black wire. If the tester doesn't light at all that means you have an open circuit or broken wire.
Jack