Define $|x|^{-2}$ be the linear functional on $C^\infty_c(\Bbb{R})$ which is defined as :
$$\phi\mapsto\int_{\Bbb{R}_+}\left(x^{-2}(\phi(x) + \phi(-x) - 2\phi(0))\right)dx$$
I have a question for this definition,it seems this linear functional is only well defined for test function $\phi(0) = 0$ only due to the last constant term and singularity $x^{-2}$ at origin.Why it's well defined linear functional for all test function? which makes it $C_c^\infty \to \Bbb{R}$ maps into?