Integral of products

I have come up with a way to find continuous products the way an integral finds continuous sums. i.e., by multiplying very  small intervals. The way to find the continuous product of a function is: e^∫[ln f(x)] dx. Input limits a and b into the integral. The ln function is undefined at 0, and a continuous product is 0 at 0, so, if a function crosses 0 between a and b, the continuous product is irrelevant. Negative numbers are a curious case, because if there are an odd number of negative terms, the function is negative, whereas if there are an even number, it is positive. Since we're taking infinitely small intervals, we can't determine whether there are an odd or even number of negative terms. One could argue that this means that there the number must be even, since it's broken up into infinite parts, therefore, the region that it is negative for will also be divisible by 1/(infinity/2). If one assumes this stance, then the correct function is e^∫ln |fx| dx, if not, put a negative sign in front, It is an interesting question, though.