The domain of a function so we can talk about limits I'm rather confused about the domain of the function that is necessary so we can talk about limits. For example here, the definition of limit considers a function $f:I\longrightarrow \mathbb{R}$ where $I$ is an open interval on $\mathbb{R}$ containing $c$. In this case the condition to talk about $\lim_{x \to c}$ is that $f$ should be defined on the interval $I$, except possibly on $c$ (This definition is the same like in Spivak's book). Now I saw in a blackboard in my school the definition considering $f:I\setminus \{c\}\longrightarrow \mathbb{R}$, and I think that this definition is wrong because in this case I couldn't talk about the limits on every point of the function since then the domain should be empty and also because $f:I\longrightarrow \mathbb{R}$ is not the same function as  $f:I\setminus\{c\}\longrightarrow \mathbb{R}$ or $f:I\setminus\{c'\}\longrightarrow \mathbb{R}$ where $c$ and $c'$ are points on the interval. Then I went to the teacher and ask him about it and he told me that this is the definition he had taken from Ethan Bloch's book and that he doesn't see any problem though and he gave me an explanation that doesn't convence me. So please, I appreciate all your comments. 
Also I don't understand why it is necessary to take $I$ as an open interval and not just any arbitrary subset of $\mathbb{R}$ containing $c$. 
EDIT: 
This is the definition as given in Ethan Bloch's book. To make precise my question let's consider the function $f:\mathbb{R}\longrightarrow \mathbb{R}$ such that $f(x)=x+1$. In this case, if I used such a definition, I cannot talk about the limit on the function $f$ at any point $c$ in $\mathbb{R}$ because the definition applies only if the function is not defined on $c$. If I applied this definition to the function $g:\mathbb{R}\setminus\{c\}\longrightarrow \mathbb{R}$ such that $\forall x(g(x)=f(x))$ and supposing the limit exists then the limit is defined on the function $g$ not on the function $f$.  
 A: 
...in this case I couldn't talk about the limits on every point of the function since then the domain should be empty...

You are correct that $f:I \setminus \{c\} \to \mathbb{R}$ and $f:I \setminus \{c'\} \to \mathbb{R}$ are different functions. What the book, and probably your teacher, is trying to convey, is that in order to take the limit at $x_0$, the function must be defined on a neighborhood $(x_0-\delta, x_0+\delta)$ of $x_0$, except that we allow for the moment that perhaps it is not defined at $x_0$. If it is defined at $x_0$, that is fine also, and still the limit may or may not exist. In fact, the definition of $f$ at $x_0$ makes no difference as to its limit. If it bothers you to take the limit of $f:I \setminus \{c\} \to \mathbb{R}$ as $x \to c$, then instead take the limit of 
$$g:I\to \mathbb{R} \\ g(x):=\begin {cases} f(x) &x\neq c \\0 & x =c\end{cases},$$
and define $\lim_{x\to c}f(x) $ as $\lim_{x\to c}g(x) $ (here zero could be replaced with any real number). 

Also I don't understand why it is necessary to take I as an open interval and not just any arbitrary subset of $\mathbb{R}$ containing $c$.

This is because the notion of a limit is supposed to hold as $x$ gets arbitrarily close to $c$. Defining it on an open set $S$ is a way of saying "For every $x\in S$, the set also contains all the points arbitrarily close to $S$". This is of course a loose way of speaking, because no single point is arbitrarily close to another point, but what we mean is that there is a neighborhood around $x$ which is contained in $S$.
