It would not be seamless but companies realize this may happen and do have plans for internet outage.When the post office can't deliver their mail and the phone company can't complete their calls, and the banks can't cash their checks how well will their "backup plans" work ?
One little corner of the internet being down is no big deal, but the whole thing going down takes *a lot* of infrastructure with it.
The phone lines will still work. People will just have to use phone and paperwork fo awhile.
A stores internal inventory system only needs internet to send the order. That can be done by fax.
The old systems will still work but may take awhile to implement.
The big problem would be people freaking out because they think the world is over.