Under what circumstances would the compressor continue to run when there's an oil failure?

The situation where a compressor continues to run despite an oil failure is typically related to the condition and functionality of the differential switch, which is designed to monitor oil levels and provide protection to the compressor. If the wire leading to the differential switch is broken, the switch cannot properly detect the oil level or pressure loss. As a result, it would fail to send a signal to shut off the compressor, allowing it to run without the necessary oil lubrication. This can lead to severe damage to the compressor due to overheating and mechanical wear, as oil is crucial for both lubrication and cooling.

In contrast, if the differential switch is closed, it indicates that the system is functioning correctly and the oil level is adequate. A functioning heater wouldn't directly impact the compressor's operation in the event of an oil failure; instead, it is more relevant to the maintenance of oil temperature and viscosity. Lastly, a faulty control would likely lead to erratic compressor behavior, but it would not specifically result in the compressor running during an oil failure if everything else was functioning correctly. Thus, broken wiring to the differential switch is the key factor that could lead to the compressor continuing to operate despite inadequate lubrication conditions.