I am reading the wikipedia articla on Dirac delta, and as far as I understand it, it is saying that only for functions with compact support $f$:
$$\int_\mathbb R \delta_t(s)f(s)ds=f(t)$$
Why the restriction? I would like to use the delta function with distributions with infinite support.
The linear map $\delta_{t}$ can act on the functions, as long as the function is defined at $t$, and the act is simply $\left<\delta_{t},f\right>=f(t)$. The problem is that, whether this linear map is continuous with respect to certain topology. For Schwartz functions $f$, not necessarily have compact support, $\delta_{t}$ is still continuous on the locally convex topology generated by semi-norms $\rho_{\alpha,\beta}(f)=\sup_{x\in\mathbb{R}^{n}}|x^{\alpha}\partial^{\beta}f(x)|$.
