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I would like to gain intuition how "singular" a distribution can be.

Let $X \in \mathbb R^n$ be open, $\mathcal D(X)$ be the smooth test functions on $X$ with canonical LF topology, and let \mathcal D'(X) be the topological dual of $D(X)$.

Let \Psi \in D'(X). Can the singular support of $ \Psi$, i.e. the complement in $X$ of the largest open subset of $X$, on which $\Psi$ is a locally integrable function, be open?

For example, the Dirac delta has singular support in $0$. Similarly, integrals with respect to Hausdorff measures on finite measure submanifolds of $X$ may be distributions, whose singular support is not open.

All examples of distributions I know constitute of a function "disturbed" by a "small" singularity.

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    May be what you want to say is whether $\mathop{singsupp} \Psi$ can *contain* a nontrivial open set?2010-12-09

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