Confusion about the definition of analytic and singularity.

In my textbook the definition of analyticity is given as

A function is said to be analytic in a domain D if f(z) is defined and differentiable at all points of D. The function f(z) is said to be analytic at a point z = $$z_0$$ in D if f(z) is analytic in a neighborhood of $$z_0$$.

Also, by an analytic function we mean a function that is analytic in some domain.

And the definition of singularity is given as

Singular point of f(z) is a point z = c at which f(z) is not analytic (but such that every disk with center c contains points at which f(z) is analytic). We also say that f(z) is singular at c or has a singularity at c.

What confuses me the most is the definition of singularity. How can it be possible that the points next to a singular point z = c are analytic when the point z = c is not differentiable or not even defined. According to the definition of analyticity, why wouldn't the non-analyticity of z = c causes the adjacent points to be non-analytic and so on.

Take, for instance, the function $$\iota\colon\mathbb{C}\setminus\{0\}\longrightarrow\mathbb C$$ defined by $$\iota(z)=\frac1z$$. It is not defined at $$0$$, right?! But you can check from the definition that is an analytic function. The fact that it is not defined at $$0$$ doesn't affect what happens around that point.
And, yes, $$\iota$$ has a singularity at $$0$$.