Singularity

Singularity is a term used across several disciplines to describe a point at which a system ceases to behave in a regular or predictable way. In mathematics, a singularity of a function is a point where the function is not analytic or not defined. Common examples include a removable singularity, where the function may be redefined to remove the problem; a pole, where the function grows without bound near the point; and an essential singularity, where the function exhibits highly non-polynomial behavior. The study of singularities is central to complex analysis and algebraic geometry, with many techniques relying on analytic continuation.

In physics, a gravitational or spacetime singularity refers to a location where curvature becomes infinite and the laws of physics as currently understood cease to apply. Such singularities are predicted by general relativity in scenarios like gravitational collapse and the early universe, and are the subject of Penrose–Hawking singularity theorems, which imply that under broad conditions, singularities arise.

In technology discourse, the term technological singularity refers to a hypothetical future point at which artificial intelligence or other technologies advance beyond human capabilities in a runaway fashion, potentially transforming society. Proponents argue it could accelerate progress dramatically; critics warn that outcomes are uncertain and difficult to predict.

Other uses include singularities in wave optics and catastrophe theory, where caustics and critical points represent breakdowns in straightforward descriptions. Overall, "singularity" denotes a break with regular patterns, whether in mathematics, physics, or speculative futurism.