= Solution
<Linearity> follows from <linearity> of differentiation and point evaluation. For a <test function> supported in a fixed compact set $K$, its value and <derivative> vanish at any point outside $K$. Therefore
$$
|\phi'(0)+\phi(1)|\leq\sup_K|\phi'|+\sup_K|\phi|\leq2\max_{0\leq j\leq1}\sup_K|\phi^{(j)}|.
$$
This proves the <continuity> estimate with $m=1$ and $C_K=2$ for every $K$. Thus the stated mapping is a <distribution> of order at most one, supported in $\{0,1\}$. In particular, a sequence tending to zero in the <test function> topology has both $\phi_n'(0)\to0$ and $\phi_n(1)\to0$.
Back to article page
