For intuition, think about the analogue for power series: if $g(z) = \sum_{n \geq 0} a_nz^n$ converges on the disc $|z| < R$ and at a point $z_0$ with $|z_0| = R$ the values $g(z)$ converge as $z \rightarrow z_0$ radially from the inside of that disc, does the series $g(z_0)$ converge? No, and the geometric series (which is an analogue of $\zeta(s)$, in the sense of being the power series with all coefficients = 1) is an example of this: $G(z) = \sum_{n \geq 0} z^n$ converges for $|z| < 1$ with value $1/(1-z)$, which is continuous for all $z \not= 0$, so when $|z_0| = 1$ and $z_0 \not= 1$ the the numbers $G(z)$ converge as $z \rightarrow z_0$ radially but the series $G(z_0)$ does not converge.

A basic result about boundary behavior for power series is Abel's theorem: if the series $g(z_0)$ *does* converge then it must be the limit of $g(z)$ as $z \rightarrow z_0$ radially (in particular, the radial limit of $g(z)$ exists). The proof of this carries over to the case of Dirichlet series: in your notation, if the series $f(s_0)$ *does* converge then it must be the limit of $f(s)$ as $s \rightarrow s_0$ from the right (and, in particular, the limit of $f(s)$ at $s_0$ from the right exists). Of course this doesn't help you to prove the Dirichlet series $f(s_0)$ actually converges; it only tells you what the value of $f(s_0)$ would have to be if that series converges.

A starting point for this circle of ideas is http://en.wikipedia.org/wiki/Abelian_and_tauberian_theorems