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.

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

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