From my understanding of distributions, the distribution, $T$, is defined as $0$ on the set $\Omega$ when $T(\phi)=0$ for all test functions $\phi$ with support $\Omega$. Further the support of the distribution is the complement of the set of points and its neighborhoods where $T$ is zero.
From this definition, I can understand how the $\delta$ and $\delta'$ distributions have the support $0$. However, I recently read from an excerpt of a textbook that the $\delta'$ distribution "hangs on at least infinitesimally, beyond its support." From what I know, both distributions are supported in a neighborhood of $0$, but what I'm struggling to understand is how $\delta'$ could be supported beyond that neighborhood. I included the excerpt below with the sentence that is confusing me highlighted.
By the definition of the derivatives of a distribution,$$\langle \delta',\phi \rangle = -\langle \delta,\phi' \rangle$$ $\delta'$ "depends on the (infinitesimal) neighborhood of 0" since the definition of $\delta'$ is dependent on the derivative of $\phi$.