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In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
For example, the real function
has a singularity at x = 0 {\displaystyle x=0} , where the numerical value of the function approaches ± ∞ {\displaystyle \pm \infty } so the function is not defined. The absolute value function g = | x | {\displaystyle g=|x|} also has a singularity at x = 0 {\displaystyle x=0} , since it is not differentiable there.
The algebraic curve defined by { : y 3 − x 2 = 0 } {\displaystyle \left\{:y^{3}-x^{2}=0\right\}} in the {\displaystyle } coordinate system has a singularity at {\displaystyle }. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.