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powerful concepts from the world of functional languages, like Scheme

Continuations are powerful concepts from the world of functional languages, like Scheme, but they are becoming popular in other languages as well.

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the objects that represent your stack trace

Basically continuations are the objects that represent your stack trace, so you can just capture a snapshot of your stack trace, save it away and then restore it and run it later, and you don't have to keep a physical thread running for that particular stack trace at all times, you can just freeze that particular stack trace into a database and restore that particular stack trace afterwards and that really allows you to scale to really any number of concurrent so to say virtual threads running at any one time.

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a feature that requires extensive (and expensive) runtime support, so extensive (and expensive)

Continuations are a feature that requires extensive (and expensive) runtime support, so extensive (and expensive)

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an important part of the language Scheme

Continuations are an important part of the language Scheme, and I invite you to learn Continuations — Continuations is a very powerful, very elegant language.

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the most basic of the various language features we are considering

Continuations are the most basic of the various language features we are considering, and the basis of the others, so let us begin there.

a feature that require the programming language they are part of, to be implemented in a way conducive to their presence

Continuations are a feature that require the programming language they are part of, to be implemented in a way conducive to their presence.

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language constructs that let you quickly store the state of a thread, and execute that thread later

As an aside, while describing continuations, Bruce Tate writes: 'Continuations are language constructs that let you quickly store the state of a thread, and execute that thread later.'

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a language feature that gives user the ability to completely control the execution control flow(first class continuation)

Continuations are also a language feature that gives user the ability to completely control the execution control flow(first class continuation).

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a means to save the current state of execution, allowing for resumption of execution at an arbitrary later date

Continuations are a means to save the current state of execution, allowing for resumption of execution at an arbitrary later date.

functions that represent 'the remaining computation to do

Continuations are functions that represent 'the remaining computation to do.'