There's two parts to this answer
1. What do you want from "monitoring"?
2. What does "continuous" mean in this context?
The REST API for JIRA is best thought of as a pull process. You have a system which asks JIRA questions or gives it commands by firing REST requests at it. This can never match the dictionary definition of "continuous" because your system is going to have to ask for information every X hours/minutes/seconds/whatever.
So, if you're happy to define your "continuous" as something like "every X minutes", you can use the REST API to do things like "list all issues updated since the time I last looked" and process them.
You will run into problems if X is too small though, as you'll be hammering the JIRA with requests
My instinct is to do it the other way around. Get JIRA to poke your system whenever the things you want to "monitor" happen.
Ok, the best way to do that is from the other end - on the "create" transition in the workflow, have a post-function that sends information over to your application.
If you have to do it from the outside, then you'll need to write something that polls JIRA with "project = X and created > 'last time I looked'" over the REST API
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