Difference between "undefined" and "does not exist"

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 $x = 0$?

Solution 1:

In the particular example you gave: The derivative is defined as $\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$ and, as it is with limits, this limit may or may not exist. In the case $f(x) = |x|$ and $x=0$ the limit just does not exist and hence, this is the right wording. On the other hand there are possible definitions of a derivative of $f(x) = |x|$ at zero (e.g. using convex analysis one may define it to be the whole interval $[-1,1]$) and hence, it seems appropriate to say that the derivative is undefined.

In general "does not exists" and "is undefined" are very different things at a practical level. The former says that there is a definition for something which does not lead to a mathematical object in a specific case. The latter says that there is just no definition for a specific case. Of course, one can interchange both formulation some times (as in you example, at least in my opinion).

Solution 2:

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 $x$.