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

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.

Real analysis

In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).

To describe the way these two types of limits are being used, suppose that

f

(

x

)

{\displaystyle f(x)}

is a function of a real argument

x

{\displaystyle x}

, and for any value of its argument, say

c

{\displaystyle c}

, then the left-handed limit,

f

(

c

−

)

{\displaystyle f(c^{-})}

, and the right-handed limit,

f

(

c

+

)

{\displaystyle f(c^{+})}

, are defined by:

f

(

c

−

)

=

lim

x

→

c

f

(

x

)

{\displaystyle f(c^{-})=\lim _{x\to c}f(x)}

, constrained by

x

<

c

{\displaystyle x<c}

and

f

(

c

+

)

=

lim

x

→

c

f

(

x

)

{\displaystyle f(c^{+})=\lim _{x\to c}f(x)}

, constrained by

x

>

c

{\displaystyle x>c}

.

The value

f

(

c

−

)

{\displaystyle f(c^{-})}

is the value that the function

f

(

x

)

{\displaystyle f(x)}

tends towards as the value

x

{\displaystyle x}

approaches

c

{\displaystyle c}

from below, and the value

f

(

c

+

)

{\displaystyle f(c^{+})}

is the value that the function

f

(

x

)

{\displaystyle f(x)}

tends towards as the value

x

{\displaystyle x}

approaches

c

{\displaystyle c}

from above, regardless of the actual value the function has at the point where

x

=

c

{\displaystyle x=c}

.