Showing $T$ is a tempered distribution

Question

Let $\mathcal{S}(\mathbb{R})$ denote the Schwartz functions on $\mathbb{R}$.

For $\phi \in \mathcal{S}(\mathbb{R})$, let

$\displaystyle{T(\phi) = \lim_{\epsilon \to 0^{+}} \int_{\epsilon}^{\infty} \frac{\phi(x) + \phi(-x) - 2\phi(0)}{x^2}dx}$.

I need help with showing that $T$ is a tempered distribution on $\mathbb{R}$.

The linearity property clearly comes via the linearity of the integral, however I am unsure how to show that $T(\phi)$ is both finite and bounded.

Answer 1

I would think boundedness comes from the mean value theorem, where $|\phi(x) - \phi(-x)| \leq 2 x \sup_y |\partial_x \phi(y)|$.