This comes from question 9.11 of Folland's Real analysis textbook. Unfortunately, I have no idea to how to start with this question. So can some one help me with part $a$?
For part $a$, I can not come up with any theorems which relate the distribution to the derivative of the test function, except the definition of derivative on distribution, which does not seems to be useful in solving this question.
BTW, the support is defined to be the complement of the maximal open subset of $R^n$ on which $F$ is $0$.
Update: I can solve part $a, b, c$ now. Can someone point out how to deal with part $d$?
I think I forgot an important fact that $<\delta,\phi>= \phi(0)$ in proving part d, and that is how the question get the $\delta$ function...