What will happen with digital money if the Internet shuts down
If you have a look globally there are countries which shut down the internet. In China for example, the government put censorship on spreading more trading with Bitcoin and other cryptocurrencies. China is becoming more and more a model to shutdown, controlling the online world. Some African countries have now also started doing it. The Democratic Republic of the Congo became the latest country who cut its people off from the Internet because the voters await the results of the weekend's presidential election.
Bitcoin and the cryptocurrencies are software around the world. It requires the internet to operate. This technology runs on a blockchain in the form of a Peer-to-Peer network. This means every computer which is connected to another operates as a server. That why the blockchain is tamper-free, secure and transparent. Everyone has access to this data container and once a transaction has been made, it is not possible to change it.
If the internet goes down or just shuts down, the blockchain won't function. If the country stops the internet or do what China does, to control the internet, the coins are still yours although you can, in a worst-case scenario, lose some although it is not really possible.
Miners will still be able to mine but they will have to do it solo and not in a team anymore. What also can happen is the time of transactions will be slower because fewer miners communication with each other.
You, as an investor in Bitcoin or cryptocurrencies, wouldn't be able to move your Bitcoins or cryptos unless the miners start or develop a radio frequency communication channel to transact with mail. This can still be a way to transact Bitcoins or Cryptos without the use of the Internet. It will be possible if you have your private keys to operate outside the country.