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Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

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