= Solution
A <distribution> on $(a,b)$ is a continuous <linear functional> on the <space of test functions> $\mathcal D(a,b)=C_c^\infty(a,b)$. It has <order of a distribution> at most $m$ if, for every <compact set> $K\subset(a,b)$, there is $C_K$ such that
$$
|\langle u,\phi\rangle|\leq C_K\max_{0\leq j\leq m}\|\phi^{(j)}\|_\infty
\qquad(\operatorname{supp}\phi\subseteq K).
$$
The <order of a distribution> is the least such $m$. When saying “of order $m$” means merely that this estimate holds, the wording allows a smaller least order.
For each such $K$, the <cutoff function> theorem supplies a nonnegative $\chi\in\mathcal D(a,b)$ equal to one near $K$. If $M=\|\phi\|_\infty$ and $\operatorname{supp}\phi\subseteq K$, then $M\chi\pm\phi\geq0$ everywhere. Positivity of the <linear functional> $V$ therefore gives
$$
0\leq M V(\chi)\pm V(\phi),\qquad
\boxed{|V(\phi)|\leq V(\chi)\|\phi\|_\infty.}
$$
Here $V(\chi)\geq0$ is finite and independent of $\phi$. This is precisely the local bound for an <order-zero distribution>, so the given <positive distribution> is continuous on the <space of test functions>. The zero <distribution> also satisfies the estimate.
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