Partial Derivatives of a continuous function is the Dirac distribution

Question

I am having trouble with the following question and a detailed solution would be immensely appreciated:

Determine a continuous function $f:\to \mathbb{R}^n \to \mathbb{F}$ and a multi-index $ \alpha \in \mathbb{N}^n$ such that $\partial^{\alpha}f = \delta_0$

Answer 1

Let $f(x) = \prod_k x_k 1_{[0,\infty)}(x_k)$ and define the distribution $T(\phi) = \int f(x) \phi(x) dx$.

Note that $\partial^{(2,...,2)} T(\phi) = \int f(x) \partial^{(2,...,2)} \phi(x) dx = \phi(0)$.