from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the Heaviside function. If g is a continuous function (or more generally a distribution) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, −f), where f is a holomorphic function on the complement of I defined by This function f jumps in value by g(x) when crossing the real axis at the point x. The formula for f follows from the previous example by writing g as the convolution of itself with the Dirac delta function. Using a partition of unity one can write any continuous function (distribution) as a locally finite sum of functions (distributions) with compact support. This can be exploited to extend the above embedding to an embedding If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then is a distribution, so when f has an essential singularity then looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.) Operations on hyperfunctions Let be any open subset. By definition is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly: The obvious restriction maps turn into a sheaf (which is in fact flabby). Multiplication with real analytic functions and differentiation are well-defined: With these definitions becomes a D-module and the embedding is a morphism of D-modules. A point is called a holomorphic point of if restricts to a real analytic function in some small neighbourhood of If are two holomorphic points, then integration is well-defined: where are arbitrary curves with The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected. Let be the space of hyperfunctions with compact support. Via the bilinear form one associates to each hyperfunction with compact support a continuous linear function on This induces an identification of the dual space, with A special case worth considering is the case of continuous functions or distributions with compact support: If one considers (or ) as a subset of via the