The Internet was originally designed to be robust and have some capability to route around damaged areas. Once the "last mile" services (DSL, cable, business-class connections, mobile wireless) are restored in many types of large-scale disasters there's a good chance that the "backbone" (connectivity beyond the last mile) will be up or will come up fairly quickly.

Presumably Internet connectivity will be unaffected outside the disaster zone, and presumably most larger services will be designed robustly with multiple datacenters in multiple geographic region.

I'm optimistic that most larger companies based in earthquake-prone areas, for example, have datacenters in other locales.