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Yes, the Ably REST API is used within the Azure Functions to publish a message. The Ably platform then uses WebSocket connections to the connected player devices to deliver the messages.
No, since the functions are short-lived it doesn't add much value to setup a WebSocket connection. You could choose for the premium tier or even a full AppService tier for Azure Functions and then you do have instances that are staying alive. But in my opinion that makes it less serverless.