I have to show that $u \in \mathscr{D}^{\prime}(\Omega)$ is not a regular distribution.
Let $\Omega = (0,1)$ Define,
$\langle u, \varphi\rangle=\sum_{j=1}^{\infty} \varphi^{(j)}\left(\frac{1}{j}\right) \quad$ for every $\quad \varphi \in \mathscr{D}(\Omega)$
So according to the definition there must be a $u$ which is locally integrable in $\Omega$, i.e. $u \in L_{l o c}^{1}(\Omega)$ in order for this to be a regular distribution. Hence, in order to prove that this is not a regular distribution I would need only show that u is not locally integrable. I think that as I take subsequent derivatives then I will have the sum tend to $0$, yet that would mean that the integral is well define and won't prove what I'm trying to show. What would a good way to start be?
Suppose there is a locally integrable function $g$ such that $\int g\phi =\sum \phi^{(j)}(\frac 1 j)$ for every test function $\phi$. Taking test functions with support in $( \frac 1 {j+1},\frac 1 j)$ we see that $f=0$ a.e. on this interval for each $j$. Also $f=o$ a.e. outside $(0,1)$. Hence $f=0$ almost everywhere leading to the contradiction $\sum \phi^{j)}(\frac 1 j)=0$ for every test function $\phi$.