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How do you say the scientific version of singularity, like the infinitely dense center of a black hole?

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Same word applies: "une singularité". You may use "une singularité gravitationnelle" when you are talking about a "trou noir".

When talking about singular functions, in a broader sense, I am trying to promote the nice French word of "affût" (in similarity to "sharp"), but without much success so far.

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