What's the time complexity bound for the Knapsack with real weights?
If you have n random real numbers, then you have 2^n possible real sums. If your knapsack has size S, you would expect 2k different sums in an interval $S-k\cdot 2^{-n}, S+k\cdot 2^{-n}]$, in other words calculating sums with a precision of $k\cdot 2^{-n}$ isn't enough to give the optimal sum. 

With random integers, you can find an optimal sum because of the $2^n$ possible sums, there will be gazillions getting the same result. You wouldn't be able to list all choices that fill the knapsack exactly because there are to many of them, but you can find one. 

You can most likely solve the problem "find items that fit in the knapsack, and have a total size at most eps less than the optimal size".