I believe this is a problem with the temperature sensor of this Kenmore. The troubleshooting is a bit involved with about 13 steps and can involve the use of a multi-meter.
How to Troubleshoot an F3 Code on a Kenmore Oven Model # 30168 (weekand.com)
Turn off electricity to the oven by turning off the circuit breaker or disconnecting the power cord from the wall outlet.
Wait for 5 minutes and restore power to the Kenmore oven; this power interruption will reset the electronic oven control module.
Turn on the heat to the oven; if the F3 error code returns, turn off the electricity to the oven again or turn off the gas supply by closing the gas **** at the oven's gas supply tube.
Wait for the oven to cool.
Open the oven door and look inside to locate the oven temperature sensor. From the front of the oven, it is usually in the upper left-hand corner. The sensor is a thin metal probe about half the diameter of a pencil that protrudes from the rear of the oven into the oven cavity.
Remove the two screws securing the sensor to the oven wall and carefully pull the sensor probe toward you until you can see the plastic wire connecting clips.
If no screws are visible, pull the oven away from the wall, remove the upper rear panel and locate the temperature probe in the upper right-hand corner. From this side, it will look like a small oblong plate with two wires projecting from it.
Remove the two screws and carefully pull out the sensor from inside of your Sears wall oven.
Disconnect any plastic clips that connect probe wires to electronic oven control (EOC).
Set your probe on a flat surface and place one lead of an ohm meter on each of metal tabs in plastic wire connector attached to probe.
At room temperature, ohm reading should read within 10 percent of 1,100 ohms.
If reading is higher or lower than that, replace your sensor.
If your sensor tests OK, check for any disconnected, broken or burnt wires in rear of range.
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