Elaborating a little on Bill_K's "It's just the way eigenstates work"...
The extension from power series to continuous functions is the subject of more sophisticated versions of the spectral theorem. Cf. Kreyszig, theorems 9.9-1 and 9.10-1.
Wikipedia
http://en.wikipedia.org/wiki/Spectral_theorem explains how, for an ordinary self-adjoint operator ##A## with spectrum ##\sigma(A)##, one can express ##A## as
$$
\def\<{\langle}
\def\>{\rangle}
A ~=~ \int_{\sigma(A)} \lambda dE_\lambda ~=~ \int_{\sigma(A)} \lambda |\lambda\>\<\lambda| ~.
$$
Unfortunately, Wiki doesn't explain extensively that the extension to a continuous real-valued function ##f## then takes the form
$$
f(A) ~=~ \int_{m-0}^M f(\lambda) dE_\lambda ~,
$$
where the integral is understood in the sense of uniform operator convergence -- cf.
http://en.wikipedia.org/wiki/Operator_topology , and ##f## is defined on ##[m,M]##, where
$$
m ~=~ \inf_{\|x\|=1} \<A\psi, \psi\> ~,~~~~~
M ~=~ \sup_{\|x\|=1} \<A\psi, \psi\> ~.
$$
(For details, one must read Kreyszig, or some other book on functional analysis.)
Anyway..., although there are still some caveats on which functions can be used with which operators, for physics purposes it's a very class.
Additional warning: there are various operator topologies (i.e., various senses in which operators can be considered equal), so one must take care when applying the more sophisticated versions of the spectral theorem.
