Trouble with fuel level meter
Although theoretically this might be a failure of either the meter or the level transmitter, the second one is much more likely and, first of all, much easier to check (to get at the meter you would need to remove and disassemble the instrument panel).
Fold the rear seat forwards, the transmitter is there under the insulator mat, covered by a circular plastic cover. Disconnect the wiring, unscrew the big black plastic nut. Gasoline is highly flammable, take the necessary precautions before you start! Lift the transmitter from the tank, let the fuel flow out from the sensor (the hole is rather small, this might take several minutes especially if the tank is full) and let it dry. Cover the hole in the tank to avoid the ingress of dirt and the danger of fire.
With the transmitter unit in your hands, shake and turn it upside down to hear if the float is sliding up and down freely. Reconnect it to the wiring, hold it horizontally and by gently tapping it with your finger, try to send the float to the other end. The meter reading should be practically proportional to the movement of the float. The warning light should come one near the end. This test reveals if the sensor is completely broken.
To find out more, drive until the warning light is coming on and the meter shows only a little amount left. Observe the meter reading carefully. Remove the sensor and submerge it into a bowl of fuel (highly flammable vapors, again, be careful!) until you get the same readout on the meter. Note the level on the sensor. Repeat this procedure when you have replenished the tank (noting the amount you could put in). Now you have two marks on the sensor and you also know the liter values they correspond to: it's quite simple to measure how the gauge answers anywhere between this extremes. Don't expect a linear relation between the amount of fuel and the meter reading, though. It is supposed to cheat by showing less fuel at the end than actually is in the tank, to help you avoid running out of fuel.
By the way, it is very important to replenish the fuel tank well before it is actually empty. It is advisable to do so as soon as the warning lamp comes on. The fuel pump is both cooled and lubricated by the fuel flowing through it. If the car becomes too dry, the fuel pump has to operate under more stress and this will shorten its life.
The transmitter is submerged in fuel forever and its float constantly scrapes the surface of the resistor track formed on the printed circuit board. After several years of use, the glue between the printing and the board wears out. The contacts for the warning light are independent from the level resistor track and they are bigger, hence the glue holds them much stronger; it is common for the level gauge to fail while the warning lamp still functions correctly. If the transmitter wears out totally, the only solution is to renew it.