Noness

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 f ( x ) = 1 / x {\displaystyle f(x)=1/x} has a singularity at x = 0 {\displaystyle x=0} , where the value of the function is not defined, as involving a division by zero. The absolute value function g ( x ) = | x | {\displaystyle g(x)=|x|} also has a singularity at x = 0 {\displaystyle x=0} , since it is not differentiable there. The algebraic curve defined by { ( x , y ) : y 3 − x 2 = 0 } {\displaystyle \left\{(x,y):y^{3}-x^{2}=0\right\}} in the ( x , y ) {\displaystyle (x,y)} coordinate system has a singularity (called a cusp) at ( 0 , 0 ) {\displaystyle (0,0)} . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.