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I am reading Stein's book on Complex Analysis, and it defines:

Definition: A function $f$ on an open set $\Omega$ is meromorphic if there exists a sequence of points $\{z_0, z_1, z_2, ... \}$ that has no limit point in $\Omega$, and such that

$(i)$ the function $f$ is holomorphic on $\Omega - \{z_0, z_1, z_2, ... \}$

$(ii)$ $f$ has poles at the points $\{z_0, z_1, z_2, ... \}$

A bit later, there is this theorem

Theorem: The meromorphic functions in the extended complex plane are the rational ones.

From the definition, it appears that meromorphic functions can have an infinite number of discontinuities. However, I believe rational functions are of the form $\dfrac {p(z)}{q(z)}$ where $p,q$ are polynomials, so they can have a maximum of $\deg q$ number of poles. How can a function with an infinite number of poles be rational?

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    $\begingroup$ The extended complex plane is compact, so there is no infinite sequence with no limit point on it $\endgroup$ – kneidell Mar 3 at 17:12
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    $\begingroup$ @kneidell Ah okay thanks! $\endgroup$ – Ovi Mar 3 at 17:14

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