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The intuition behind continuations

A continuation of the evaluation of an expression E in a surrounding context C represents the future of the computation, which waits for, and depends on, the value of E

 

Context C and expression E

 Intuitive continuation of E in C

(+ 5 (* 4 3))
 The adding of 5 to the value of E
(cons 1 (cons 2 (cons 3 '())))
 The consing of 3, 2 and 1 to the value of E
(define x 5)
(if (= 0 x)
    'undefined
    (remainder (* (+ x 1) (- x 1)) x))
 The multiplication of E by x - 1 followed by a division by x