This night the battery saving function turned on automatically (battery level < 15%) which is the way I prefer it to be. I had set a vibrate only alarm as wake up alarm. The bad thing is that the alarm went off completely silent without vibration and I did not wake up. The vibrations are turned off when battery saving function is active. This must be a bug but the Sony support told me that it is not and that they can't be responisble for Google designing Android this way.
Sony's Stamina mode did not behave like this. If vibration is chosen for such an important event as an alarm then it should vibrate or at least play a sound.
Bad feature and also in general a little bad attitude by the Sony support person.
