The battery runs so long that you don't have to worry about it. Besides a portable device being used with a stationary power supply (AC-power) is a contradiction in terms
. It's far far far away from what a cellphone is
.
The screen needs power only when redrawn - therefore it remains with the last image even if the device is shut down or locked or defective. One cannot judge the status of the Sony based solely on what it displays - again very different from any other type of device (cellphone, laptop etc).