Every distribution $t\in\mathcal{D}'(\mathbb{R}^n)$ of compact support is of the form
$$ t(x) = \sum_{|\alpha|<m}\partial^\alpha t_\alpha(x), $$ where $\alpha$ is multi-index, $m$ - some constant and $t_\alpha \in C(\mathbb{R}^n)$.
It is not true that every distribution $t\in\mathcal{D}'(\mathbb{R}^n)$ is of the above form (the sum may be infinite).
However, I suppose that every Schwarz distribution $t\in\mathcal{S}(\mathbb{R}^n)$ is of the above form, i.e. $$ t(x) = \sum_{|\alpha|<m}\partial^\alpha t_\alpha(x), $$ where $\alpha$ is multi-index, $m$ - some constant and $t_\alpha \in C(\mathbb{R}^n)$. Am I right? Can you provide a reference.