The O2 sensor (at least the type used on yours) is essentially a 'binary' device, and should not stay at one voltage level for any length of time - it should toggle between roughly 0.1 and 0.9 volts. The ideal fuel/air ratio is indicated by 0.45V reading at the sensor. But since the engine management is not capable of maintaining that perfect mixture for any period of time, the mixture will constantly pass between rich and lean, with the long-term average O2 reading being 0.45V. Depending on the response of the meter you are using, this may be hard to see; it might just look like the voltage is bouncing around.

You can not really test 'continuity' of the O2 sensor; it is a high-impedance voltage-generating device, and not a resistance-variable device (like a temperature sensor), so you can not use an ohmeter to test the device itself. But regardless of what voltage appears on the sensor output, the same voltage should be present at the ECU connector as well.

Also keep in mind the ECU does not go into closed-loop mode until the engine is warm, so the O2 sensor will likely be stuck at the high- or low-level when the engine is cold. When the engine (and exhause and O2 sensor) are warm, the ECU accepts the O2 sensor as valid and uses it to fine-adjust the mixture. But again, the O2 output should not stay at 0.57V as you indicated.

Hope that helps