Editing Singularity (mathematics) (section)

{{short description|Point where a function, a curve or another mathematical object does not behave regularly}}
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 [[derivative|differentiability]] or [[Analyticity of holomorphic functions|analyticity]].<ref name=":1">{{Cite web|url=http://mathfaculty.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html|title=Singularities, Zeros, and Poles|website=mathfaculty.fullerton.edu|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.britannica.com/topic/singularity-complex-functions|title=Singularity {{!}} complex functions|website=Encyclopedia Britannica|language=en|access-date=2019-12-12}}</ref><ref name=mathworld/>

For example, the [[reciprocal function]] <math>f(x) = 1/x</math> has a singularity at <math>x = 0</math>, where the value of the [[function (mathematics)|function]] is not defined, as involving a [[division by zero]]. The [[absolute value]] function <math>g(x) = |x|</math> also has a singularity at <math>x = 0</math>, since it is not [[Differentiable function|differentiable]] there.<ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref>

The [[algebraic curve]] defined by <math>\left\{ (x,y):y^3-x^2 = 0 \right\}</math> in the <math>(x, y)</math> coordinate system has a singularity (called a [[cusp (singularity)|cusp]]) at <math>(0, 0)</math>. For singularities in [[algebraic geometry]], see [[singular point of an algebraic variety]]. For singularities in [[differential geometry]], see [[singularity theory]].