In it’s standard configuration, no. This is for safety reasons. If there was a powercut, and the batteries were to continue to feed power, they could potentially feed power back to the grid. If someone was working on the grid to restore power, they would be unaware the cables were live and it could be fatal. As a result, the battery will shut down to ensure the safety of grid operators.
However it is possible set up an Emergency Power Supply (EPS). With an EPS system, the battery will isolate your electric network, separating yourself from the grid. This can then power your home electricity network from your solar production and any stored battery power. There are limits to how much power it can provide at one time – we recommend only a few essential circuits are wired to the EPS (lights, fridge, etc), as a high current device could overload the battery causing it to shut down.