Here is the inspiration for my question: it is easy to see that the self-dual action of an element $f \in L^2(a,b)$ on $L^2(a,b) \cap C^\infty (a,b)$ does not depend on pointwise values. Indeed, if $h \in L^2(a,b) \cap C^\infty (a,b)$, then for any point $c \in (a,b)$ we can write $h = (h - h(c) \phi_{(c-\frac{1}{n} , c+\frac{1}{n})} ) + h(c)\phi_{(c-\frac{1}{n} , c+\frac{1}{n})} $, where $\phi_{(c-\frac{1}{n} , c+\frac{1}{n})}$ is a bump function with $\phi_{(c-\frac{1}{n} , c+\frac{1}{n})}(c)=1$, and since $\int_a^b f \phi_{(c-\frac{1}{n} , c+\frac{1}{n})} \, dx \to 0$ as $n \to \infty$, we have $\int_a^b f \left( h - h(c)\phi_{(c-\frac{1}{n} , c+\frac{1}{n})} \right) \, dx \to \int_a^b fh \, dx$ .
However, what if we consider instead $f \in H^{-1/2}(a,b)$ acting on $H^{1/2}(a,b) \cap C^\infty (a,b)$? To attempt a parallel of the argument above, we would need an analog of the bump functions that would work in $H^{1/2}(a,b)$; that is, we would need a family of functions $\psi_n \in C^\infty(a,b)$ that satisfy $\psi_n(c)=1$, $\|\psi_n \|_{H^{1/2}} \to 0$ as $n \to \infty$.
My question: Does such a sequence exist?
Some context as to why I'm interested: I am interested in extending weakly defined partial differential equations from half-spaces to whole spaces, and want to know when it is possible to break test functions up across the boundary.
Thanks for any help and insights!