The singularity spectrum is a function used in multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same Hölder exponent. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function.

More formally, the singularity spectrum D ( α ) {\displaystyle D(\alpha )} of a function, f ( x ) {\displaystyle f(x)} , is defined as:

D ( α ) = D F { x , α ( x ) = α } {\displaystyle D(\alpha )=D_{F}\{x,\alpha (x)=\alpha \}}

Where α ( x ) {\displaystyle \alpha (x)} is the function describing the Hölder exponent, α ( x ) {\displaystyle \alpha (x)} of f ( x ) {\displaystyle f(x)} at the point x {\displaystyle x} . D F { ⋅ } {\displaystyle D_{F}\{\cdot \}} is the Hausdorff dimension of a point set.