Gas oven electric igniter?

Your stoves igniter provides a way to light the gas PLUS it allows for the ovens safety valve to open up. As the igniter gets hotter, its resistance drops. That allows the bi-metal within the gas safety valve to open and let that gas flow out to the burner. So Yes the ignitor must stay on till the oven reaches temp setting and then it goes out when the power is removed from the circuit by the thermostat. Tip: if your oven should fail to shut down by some failure, simply unplugging the unit will kill the oven but not the top burners. Here is a short video that explains how an oven igniter works:

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Gas Oven Igniter

The igniter on my oven stays on all the time when the burner is in use too. I think the odor will pass after you run the over and burn off what ever become deposited while cleaning.

The igniter may also work as a thermo coupler. It stays in the flam and the flame heats it red. This tells the stove that the flame is lit and continue to supply gas. If the flame goes out it clues down and tells the stove no flame so turn off gas. You should be able to find a online repair manuel and it will answer this question.

Oven Ignitor

In addition to the igniter that the previous person recommended there is a relay or device that allows the gas valve to open. We did some trouble shooting using a volt meter and the Sears online parts diagram. It was cheaper to replace the valve than the igniter and it turned out to be the problem. Good Luck

Yes the ignitor stays on while the burner is lit.