It's usual in Complex Analysis see some points of the domain of a function as undefined. See for example as Wikipedia defines a removable singularity:

I've seen functions with undefined points in real calculus books as well (they call these points as discontinuities).

I don't feel very well with such points, the definition of functions says clearly that points should be defined in its domain and functions with undefined points break this rule which is unacceptable mathematics speaking.

So where am I wrong?

  • $\begingroup$ Once you remove the singularity, the function is no longer undefined at that point. It simply becomes a piecewise defined function $$\begin{align}\operatorname{sinc}(z)=\begin{cases}\sin(z)/z & \quad z \ne 0 \\ 1 & \quad z = 0\end{cases}\end{align}$$ whose domain is the entire $\mathbb{C}$. $\endgroup$
    – dxiv
    Nov 7 '16 at 5:32
  • $\begingroup$ A function is defined on a domain. If that domain does not contain a particular point, then the function is not defined at that point. There is nothing wrong with that. The positive square root of a non negative number is defined on $[0, \infty)$, and this domain does not include $-1$. When one talks about removable singularities, one is talking about extending the domain (hence defining an extension of the original function). $\endgroup$
    – copper.hat
    Nov 7 '16 at 5:38
  • $\begingroup$ @dxiv you didn't follow me. The authors treat $\sin z/z$ as a function before to define $\sin z/z=1$ at $z=0$. $\endgroup$
    – user42912
    Nov 7 '16 at 5:43
  • $\begingroup$ @user42912 If I didn't follow you, then maybe you didn't post enough context. My comment just explained the common way a "removable singularity" is understood. $\endgroup$
    – dxiv
    Nov 7 '16 at 5:49

The definition of a singularity is as follows: Let $D$ be an open set in $ \mathbb C$, let $z_0 \in D$ and suppose that $f$ is a holomorphic function on $ D \setminus \{z_0\}$. Then $z_0$ is called an isolated singularity of $f$.

Observe that $f$ is defined on $ D \setminus \{z_0\}$ !

The singularity is called removable , if there is a holomorphic $g$ on $D$ such that

$ f=g$ on $ D \setminus \{z_0\}$


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