# Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions.

• How about $\phi\mapsto\lim_{\epsilon\to 0^+}\left(\int_{-\infty}^{-\epsilon}\frac{\phi(x)}{x}\,\mathrm dx+\int_{\epsilon}^{\infty}\frac{\phi(x)}{x}\,\mathrm dx\right)$? Apr 18 '15 at 13:02
• I think the first derivative of the so-called Minkowski's Question Mark function may be one such example. Apr 18 '15 at 13:05

Joelafrite made a good suggestion: consider the first distributional derivative of a function $f$ that is not absolutely continuous. By definition, this distribution acts as $$\phi\to -\int f\phi'$$
• If $f$ is increasing (like Cantor staircase and Minkowski's ?-function), then the distribution $f'$ is a measure.
• If $f$ has bounded variation, then $f'$ is a signed measure.
• For more extreme examples, such as the Weierstrass nowhere differentiable function, $f'$ is a distribution that is hard to imagine or compare to any familiar object.
As for classification: a derivative is of order at most $k$ if its value on test function $\phi$ is bounded by $C^k$ norm of $\phi$. So, order $0$ distributions are signed measures. There are distributions of every finite order, as well as of infinite order.
• Thanks to your answer, I learned about the Minkowski's $?(x)$. Apr 18 '15 at 20:29