TITLE: Difference between "undefined" and "does not exist"
QUESTION [16 upvotes]: What is the difference between the terms "undefined" and "does not exist", especially in the context of differential calculus?  
Most calculus materials state, for example, that $\frac{d}{dx}{|x|}$ does not exist at $x = 0$.  Why don't we say that the derivative is undefined at $x = 0$?

REPLY [6 votes]: There is a little subtlety not addressed so far: 
Given a function $f:\ A\to\Bbb R$ on an open set $A\subset\Bbb R$ and a point $x\in A$, you can ask whether $f$ is differentiable at $x$. The function $f$ is differentiable at $x$ (or: has a derivative at $x$) if the limit
$$\lim_{h\to0}{f(x+h)-f(x)\over h}\tag{1}$$
exists and is finite. This limit is  called the derivative of $f$ at $x$ and is denoted by $f'(x)$.
The set $A'$ of all $x\in A$ where the limit $(1)$ exists, is the domain of a new function $f'$ associated to $f$. This  new function is called the derivative of $f$. 
When $x\in A'$ then we say that $f'$ is defined at $x$.