Daniel Velleman has written Calculus: A Rigorous First Course.
It seems to me that it's questionable whether such a course should be used for all but a small number of students, not just because most will never appreciate logical rigor, but because there's so much stuff they may appreciate that you won't be telling them about because you'll be explaining logical rigor instead. However, skip this paragraph and resume reading below.
Velleman introduces an unconventional notation: $f(x)\to 6$ as $x\to2^\ne.$ The superscript $\text{“}\ne\text{''}$ serves as a reminder that although $f(x)$ is not forbidden to be equal to $6$ as it is approaching $6,$ nonetheless $x$ is forbidden to be equal to $2$ as it is approaching $2.$
Besides understanding concepts in a logically rigorous way (as in $\varepsilon$-$\delta$ arguments and some other occasions), it seems to me the principal place where this is useful is in understanding why a proof of the chain rule is problematic in a way in which proofs of other differentiation rules are not. \begin{align} \frac{dy}{dx} = {} & \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \cdots\overbrace{\cdots\cdots\cdots\cdots\cdots}^{\large\text{What goes here?}}\cdots = \frac{dy}{du} \cdot \frac{du}{dx}. \\[12pt] & \lim_{\Delta x\to0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x\to0} \frac{\Delta y}{\Delta u}\cdot\frac{\Delta u}{\Delta x} = \text{?} \end{align} We have $\Delta x\to0^\ne.$ And so $\Delta u\to0.$ (And if you want to be logically rigorous, here we have relied on a theorem that says differentiable functions are continuous.)
But we have only $\Delta u\to0,$ not $\Delta u\to0^\ne,$ and that is the difficulty to be overcome.
So my question is whether, besides the proof of the chain rule, there is any other place in the conventional less-than-rigorous calculus course where Velleman's novel notation is so useful for efficiently stating the point? (Addendum in response to a posted answer: I'd have thought the following was obvious, but I guess it wasn't: I was not proposing that this helps in actually proving the chain rule.)
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$\begingroup$Here is another place where the notation is important: If $\lim_{x \to a} f(x) = b$ and $\lim_{u \to b} g(u) = c$, does it follow that $\lim_{x \to a} g(f(x)) = c$? Of course, the answer is no, but why? Here's an answer using the new notation: Let $u = f(x)$ and $y = g(u) = g(f(x))$. Then we have: as $x \to a^\ne$, $u \to b$, and as $u \to b^\ne$, $y \to c$. But the mismatch between $u \to b$ and $u \to b^\ne$ means that these cannot be combined to conclude that as $x \to a^\ne$, $y \to c$. If $g$ is continuous at $b$, then we can say that as $u \to b$ (no superscript), $y \to c$, and then we can conclude that as $x \to a^\ne$, $y \to c$.
Most calculus books have a theorem about limits of sums, differences, products, and quotients, but no systematic way to work out limits of compositions. The reason, in my view, is that they don't have the notation necessary for talking about limits of compositions. This new notation fills this gap. The new notation is used in a number of places throughout the book. (Similar notation even comes up in the proof of the fundamental theorem of calculus.)
The use of a superscript is consistent with other standard notations that use a superscript to indicate how a variable approaches a limit, such as $x \to a^+$.
In response to the second paragraph of the original post: I think the comments in this paragraph apply to a book like Spivak, but not my book. Spivak covers a lot of topics that belong in an analysis course, and as a result there are lots of standard calculus topics that he doesn't cover. But that's not true of my book, which is really a calculus book and not an analysis book. The arrow notation I have introduced is no more complicated or difficult to use than standard arrow notation, but it makes informal (non $\epsilon$-$\delta$) reasoning about limits more reliable. Isn't that what good calculus notation should do?
$\endgroup$ 15 $\begingroup$Note that in the French tradition limits of functions are not defined the same way as in America: If the function $f$ happens to be defined at the point $a$ (in most cases it is not) then according to the French the $\lim_{x\to a} f(x)$ only exists, if $f$ is continuous at $a$, whereas according to Baby Rudin the value $f(a)$ is not taken into consideration for the $\lim_{x\to a} f(x)$. The following Wikipedia entries show this difference:
Now Vellemann wants to make this distinction visible in his limit notation. For my part I'm less than happy with his proposal, because the $\ne$ command does not affect (or operate on) the constant $2$ in any way. Instead the command concerns the freedom the point $x$ has when approximating $2$. Therefore I'd write $x\to_{\ne}2$ (or similar) instead.
As an aside: I don't think that the special notation introduced here is helping the "Calculus 101" proof of the chain rule. The correct procedure is to avoid denominators in this proof altogether. See my answer to this question.
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