The first time you setup Continuum you will see this video that explains the concepts and operation of the feature.
Adjusting the monitor display to fill the screen is one of the initial steps in setting up Continuum.
You can see the Lumia 950 plugged into the Microsoft Display Dock and Tweetium running on the phone being displayed on the monitor.
Since Tweetium is a UWP it runs full screen on the external monitor in Continuum mode.
As I prepared to do the screenshots for this gallery I plugged up the 950 and found a great number of tabs open in Microsoft Edge. They were not causing any issues but just shows that tabs opened on the phone stick around for a while.
A clean view of Microsoft Edge which is basically the desktop UI.
Sports app
Action Center
Settings app looks very familiar
Desktop in Continuum mode showing multiple apps open and on the taskbar.
Tweetium for Windows
News app
Facebook (Beta)
Facebook (Beta)
SuperSite for Windows page on Facebook (Beta)
Facebook (Beta) Settings Overview
Facebook (Beta) Notifications
Outlook Calendar
Outlook Mail
Outlook Mail Settings menu
Outlook Mail Account Settings
Outlook Mail Light theme
Groove Music
Start Menu in Continuum mode. Take note of the dimmed tiles - those are the non UWP apps which will not run on the external monitor when in Continuum mode.
Another example of the dimmed tiles that indicate an app is not UWP and will not run in Continuum mode on the external monitor.