If I lose internet, shouldn't Hub still let me control stuff in my wifi?

[JDRoberts]

At present, SmartThings offers very little off-line processing, although they have said they would like to do more in the future.

Right now, the only code that runs local is SmartLighting and some parts of smart home monitor. And then only if all of the devices in that particular automation are also eligible to run locally, at least for a few hours until the hub battery dies. And even then the mobile app on your phone does not run locally. Your switches probably will run locally.

See the following thread for more details:

Local Processing in Hub V2

Continuing the discussion from What's the deal with Nest?: Correct. I was answering this question below in the global sense. i.e., There will always be the possibility of various benefits available to Official Device Handlers and SmartApps that are not available to un-certified Community Developed ones. At the moment, the local processing benefit is inapplicable to Nest in either case. I’m not sure ST has guaranteed that community developed SmartApps/Device Types will be included in local processing right after certification. Yes, it may be included down the road, and this is not yet clear on how or what the process is to enable community developed stuffs to run locally.