From Complex Variables with Applications (Orloff):
Poles refer to isolated singularities. So, we suppose $ f(z) $ is analytic on $ 0 < |z - z_0| < r $ has Laurent series $$ f(z) = \sum_{n=1}^{\infty} \frac{b_n}{(z - z_0)^n} + \sum_{n=0}^{\infty} a_n (z - z_0)^n. $$
Then it defines:
If all the $ b_n $ are $ 0$, then $ z_0$ is called a removable singularity. That is, if we define $f(z_0) = a_0$ then $f$ is analytic on the disk $|z - z_0|< r $.
I am unable to understand this definition of removable singularity. See this $ f(z) $ for example:
$$ f(z) = 1 + 2(z - 10) + 3(z - 10)^2 $$
All coefficients $ b_n $ are $ 0 $ in the above function. By the definition above, we should call $ 10 $ to be a removable singularity. But why is $ 10 $ a singularity? The function $ f(10) = 1 $ which is a finite value. The function $ f(z) $ appears to be analytic at $ z = 10 $. Then why should we call $ z = 10 $ a removable singularity here.
Is the quoted definition of "removable singularity" correct? Am I misunderstanding the definition?