is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking, ( \ 0): however, this function can be extended to a distribution on the whole as a Cauchy principal value. The preceding example raises a question: does every function which is locally integrable in ⊊ admit an extension to the whole as a distribution? The answer is negative, and a counterexample is provided by the following function: does not define any distribution on . The following example, similar to the preceding one, is a function belonging to ( \ 0) which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular coefficients: where and are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order Again it does not define any distribution on the whole , if or are not zero: the only distributional global solution of such equation is therefore the zero distribution, and this shows how, in this branch of the theory of differential equations, the methods of the theory of distributions cannot be expected to have the same success achieved in other branches of the same theory, notably in the theory of linear differential equations with constant coefficients. Applications Locally integrable functions play a prominent role in distribution theory and they occur in the definition of various classes of functions and function spaces, like functions of bounded variation. Moreover, they appear in the Radon–Nikodym theorem by characterizing the absolutely continuous part of every measure. See also Compact set Distribution (mathematics) Lebesgue's density theorem Lebesgue differentiation theorem Lebesgue integral Lp space Notes References . Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive. . Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals. . (available also as ). (available also as ). . . . . . English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically