\[ex:deltas\], which for $f\in C^\infty_c({\mathbb{R}}^2)$ perhaps amounts to $\partial f/\partial x|_{(x,t)=(0,0)}$. The important point, in any case, is that if we considered the derivative with respect to $t$ as a distributional one, we would need to be taking test functions that depended on that variable, and we do not do this; note that in Example \[ex:deltas\] the function $f$ depends only on the real variable corresponding to $P$ and not on $t$.

\[ex:deformationsdelta\] Let again $P={\mathbb{R}}$, and let us use Theorem \[thm:families\] to determine the ways in which the Dirac delta at 0, $\delta_0$, can be deformed. The condition in Theorem \[thm:families\] requires us to look for distributions $\eta$ such that $\langle\eta,f\rangle{\geqslant}0$ for all $f\in C^\infty_c({\mathbb{R}})$ such that $f(x){\geqslant}0$ for all $x\in {\mathbb{R}}$ and $f(0)=0$. Since the Taylor expansion of any such function $f$ starts at the quadratic term, which must have a nonnegative coefficient, we conclude that $\eta$ must be of the form $$\eta=\nu+a\,\partial_x\delta_0+b\,\partial_x^2\delta_0$$ for a positive measure $\nu$ on ${\mathbb{R}}$, a real number $a$, and a positive number $b>0$.

\[ex:probdeformationsdelta\] Going back to the situation of Example \[ex:deformationsdelta\], we now use the second part of Theorem \[thm:families\] to determine the ways in which the distribution $\delta_0$ can be 