Singularity (mathematics)

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.^[1]^[2]^[3]

For example, the reciprocal function $f(x)=1/x$ has a singularity at $x=0$ , where the value of the function is not defined, as involving a division by zero. The absolute value function $g(x)=|x|$ also has a singularity at $x=0$ , since it is not differentiable there.^[4]

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

^ "Singularities, Zeros, and Poles". mathfaculty.fullerton.edu. Retrieved 2019-12-12.
^ "Singularity | complex functions". Encyclopedia Britannica. Retrieved 2019-12-12.
^ Cite error: The named reference mathworld was invoked but never defined (see the help page).
^ Berresford, Geoffrey C.; Rockett, Andrew M. (2015). Applied Calculus. Cengage Learning. p. 151. ISBN 978-1-305-46505-3.

[:1-1] "Singularities, Zeros, and Poles". mathfaculty.fullerton.edu. Retrieved 2019-12-12.

[2] "Singularity | complex functions". Encyclopedia Britannica. Retrieved 2019-12-12.

[mathworld-3] Cite error: The named reference mathworld was invoked but never defined (see the help page).

[4] Berresford, Geoffrey C.; Rockett, Andrew M. (2015). Applied Calculus. Cengage Learning. p. 151. ISBN 978-1-305-46505-3.

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