The space of locally integrable functions embeds in the space of (standard) distributions. Additionally, every distribution can be thought of as a multiple derivative of some continuous function.
Does this imply that every distribution can be thought of as the sum of a locally integrable function, and a countable sum of translations of derivatives of the Dirac delta distribution?
In other words, loosely thinking, does every distribution have a "function" part, and a "delta" part?
I would like to think that this is what's implied by the theorem that every distribution is the n'th derivative of a continuous function -- in other words, that the "delta part" of any derivative will appear at the parts where the continuous function isn't differentiable. But what about functions like the Weierstrass function, which is continuous yet nowhere differentiable?