Does the camera still record if the internet goes down?

If a connected camera loses its Internet connection (either over WiFi or ethernet), it will continue to record.  During this time though, no new events (motion, human event, etc) will be shown on the seek bar.  Once the camera comes back online, all missing events will be populated and the video will be remotely accessible again.

If you have Cloud Archiving enabled for the camera, you can view past footage online while the camera is offline.  Furthermore, once the camera comes back online, the missing footage will automatically be uploaded to the cloud.