In mathematics a singularity is, informally, a place where trends break down. Suppose you have a smooth function *f(x)* that you cannot define directly, but you know in as much detail as you like how it behaves near x=0. If it goes infinite at x=2, you cannot use your knowledge around x=0 to project it beyond x=2.
The technological singularity (as I understand) was so named circa 1990 because various trends looked like getting so weird around 2035 that we cannot project them beyond that time.
(small EDIT for more clarity)
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