# Support (mathematics)

Real-valued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups.^{[7]}

In probability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra, rather than on a topological space.

Note that the word *support* can refer to the logarithm of the likelihood of a probability density function.^{[9]}

In Fourier analysis in particular, it is interesting to study the **
singular support** of a distribution. This has the intuitive interpretation as the set of points at which a distribution

*fails to be a smooth function*.

For distributions in several variables, singular supports allow one to define *wave front sets* and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).