# 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 $x = 0$?

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## 3 Answers

In the particular example you gave: The derivative is defined as a limit $\lim_{h\to 0} (f(x+h) - f(x))/h$ 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).

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I've changed the example to $\frac{d}{dx}(\frac{1}{x})$. – bwDraco Feb 6 '13 at 18:48
Why?$\mbox{}\mbox{}$ – Dirk Feb 6 '13 at 19:17
Alright, I've rolled it back. – bwDraco Feb 6 '13 at 20:31
Indeed, editing the question such that right answers get wrong is not always a good idea. If you ask for the function $f(x) = 1/x$ and whether the derivative exists or is undefined at $x=0$, then the answer would be (in my understanding) that the derivative is undefined (because its definition involves the term $f(0)$ which is undefined). – Dirk Feb 7 '13 at 8:16
By your saying that "there is a definition for something which does not lead to a mathematical object in a specific case", do you mean that the result is defined, but is something like $\infty$? – bwDraco Feb 8 '13 at 21:20

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 $x$.

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As I understand it, "undefined" means that the answer could be any real number, whereas "does not exist" means that the answer couldn't be any real number. Thus, 0/0 is undefined (since any number times zero equals zero), but 3/0 does not exist (since no number times zero equals three).

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