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I am familiar with the idea of metrics in software and I know something about using distance functions in clustering, but I don't really know what is meant by "formal distance metric". What is the difference between a distance function and a formal distance metric?

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What they probably meant is that it strictly meets the definition of a metric, as defined here. A function that is not non-negative would not pass muster, for example.

If this does not make sense, please cite the reference where you encountered the term.

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