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Continuous function: If I were to take a zero alpha cut of a Gaussian membership function centered at +2, then I would be approaching zero on the tail ends at +/- infinity. Would I claim the centroid to be at +2, 0 or undefined based on the continuous nature of the curve and the fact that there is no area under the alpha-cut at zero?

Non-continuous function: If I were to take a zero alpha-cut of a trapezoidal membership function with points {0,1,3,4}, would I claim the centroid to be at +2, 0 or undefined? In this case, I know the terminal points of the curve* are at 0 and 4, but I still have the dilemma of zero area.

*I know it's not a curve, just relating it the example above.

I feel as though I have a good heuristic grasp on what's happening here, yet I'm lacking rigorous mathematical understanding and therefore can't figure out what value needs to be assigned to the result of this calculation. Long story short, I'm writing a computer program, and I need to know whether it makes more sense to return a zero, a "reasonable" value, or to simply throw an error.

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Assuming A is a fuzzy subset of X, the zero alpha-cut is X. This leads me to believe the answer I was looking for is best labelled as undefined.

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