Past exam of the differential equations and applications course of the University of Oxford / 2010 / A / 1 / ii / b

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 $K$. ThereforeThis 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 ${\varphi }_n^{\prime }(0)\to 0$ and ${\varphi }_n(1)\to 0$.