How to continue ps4 download in background - consider, that
Automatic Downloads
To enable automatic downloads while your PS4™ system is turned on or in rest mode, select 
(Settings) > [System] > [Automatic Downloads].
Featured Content
Your PS4™ system automatically adds recommended content from PlayStation™Store to the content area. Select the added content to start the download.
System Software Update Files
Your PS4™ system automatically downloads and installs system software update files. To enable your system to automatically install files even when the update requires it to restart, select the checkbox for [Allow Restart].
Application Update Files
Your PS4™ system automatically downloads update files for games and other applications.
- To download while in rest mode, select (Settings) > [Power Save Settings] > [Set Features Available in Rest Mode] and then select the checkbox for [Stay Connected to the Internet].
- To manually download an application update, press the OPTIONS button while the application is highlighted in the content area, and then select [Check for Update].
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