Try something simpler to see how this works. For example, here’s a version of a list-sum function that receives a continuation argument (which is often called k):
(define (list-sum l k)
(if (null? l)
???
(list-sum (cdr l) ???)))
The basic pattern is there, and the missing parts are where the interesting things happen. The continuation argument is a function that expects to receive the result — so if the list is null, it’s clear that we should send it 0, since that is the sum:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) ???)))
Now, when the list is not null, we call the function recursively with the list’s tail (in other words, this is an iteration), but the question is what should the continuation be. Doing this:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) k)))
is clearly wrong — it means that k will eventually receive the the sum of (cdr l) instead of all of l. Instead, use a new function there, which will sum up the first element of l too along with the value that it receives:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (lambda (sum) (+ (car l) sum)))))
This is getting closer, but still wrong. But it’s a good point to think about how things are working — we’re calling list-sum with a continuation that will itself receive the overall sum, and add the first item we see now to it. The missing part is evident in the fact that we’re ignoring k. What we need is to compose k with this function — so we do the same sum operation, then send the result to k:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (compose k (lambda (s) (+ s (car l)))))))
which is finally working. (BTW, remember that each of these lambda functions has its own “copy” of l.) You can try this with:
(list-sum '(1 2 3 4) (lambda (x) x))
And finally note that this is the same as:
(define (list-sum l k)
(if (null? l)
(k 0)
(list-sum (cdr l) (lambda (s) (k (+ s (car l)))))))
if you make the composition explicit.
(You can also use this code in the intermediate+lambda student language, and click the stepper button to see how the evaluation proceeds — this will take a while to go over, but you’ll see how the continuation functions get nested, each with it’s own view of the list.)