I am getting repetitious so this will be my final post.

There are no rules about informal or private thinking. Language, however, and all forms of public communication are extra individual; they rely on a shared understanding of what symbols mean.

Mathematicians have developed the least ambiguous system of symbolic representation. Part of mathematical education involves teaching (1) that system and (2) the need for very careful definition and strict logical thinking.

$f(x)$ by itself is an undefined term and is a meaningless concatenation of meaningless shapes without extra individual conventions. When teaching French you don't permit students to say "le dog" as a translation of "the dog," but insist on "le chien."

If I specify $f(x) = \dfrac{1}{x}\ if\ x \ne 0$, then $f(0)$

is an undefined term without any known meaning. It may be 2 or perhaps $sin(x)$. It can be anything. There is nothing to be said about something that is not even defined.

I hate standard analysis. I find it horribly ugly. I may find it personally

convenient to remember the chain rule as $\dfrac{dy}{dx} = \dfrac{dy}{\cancel{du}} * \dfrac{\cancel{du}}{dx}$,

but I would not use it in a math class on standard analysis because those symbols have been defined as operators, not fractions.