Implementing Haskell#break
The first [] case is unnecessary, since break'' already handles empty lists.

g is unnecessary, since f is in scope.

break'' has two nearly identical base cases, which could be combined into one (you may need to use head and tail instead of pattern-matching). That's a line shorter but IMO no clearer.

My previous advice to reverse the accumulator contained a subtle flaw: it needs to find the boundary before it returns anything, so it's not lazy enough. (This is a subtlety of lazy data structures; if you don't understand it yet, don't worry about it.) This occasionally matters in cases like this:

let (a, b) = break (==' ') $ "abc" ++ undefined in head a


The actual implementation avoids this by simply adding x to the result of the recursive call, without using an accumulator:

break _ xs@[]           =  (xs, xs)
break p xs@(x:xs')
           | p x        =  ([],xs)
           | otherwise  =  let (ys,zs) = break p xs' in (x:ys,zs)