1

I'm noticing a pattern in my writing where I refer to “the set of points in $X$ such that $f$ is differentiable.” The same set appears frequently in literature, usually by way “suppose $f$ is differentiable at $x$ [$\ldots$] it follows that [result] holds wherever $f$ is differentiable,” or something similar.

Given the sheer number of properties that begin and end “where $f$ is differentiable,” I’d be surprised that there isn’t at least a common term for this set (compare kernel, zero set, support, etc.)

Or maybe there is and I just failed to learn basic maths in junior high school. Either way, it would be really convenient to have an established shorthand for these sets.

Rócherz
  • 3,976
R. Burton
  • 4,942
  • 1
    How about "the domain of $f'$"? – Robert Israel Oct 12 '22 at 04:47
  • I don't know of one, but if I'm allowed to coin a phrase, how about its "domain of differentiability"? Or shorter, "differentiable set/domain" – Akiva Weinberger Oct 12 '22 at 05:33
  • basic maths in junior high school --- I suspect "junior high school" means something different to you than to me. We had not even reached beginning school algebra (e.g. solving an equation such as $2x + 5 = 15$) when I was in junior high school, although at the time I knew a little algebra from studying some books in the school and county libraries. Someone learning calculus in junior high school should surely be ready for graduate courses in topology and measure theory by the time they begin college! – Dave L. Renfro Oct 12 '22 at 10:38

1 Answers1

3

I've seen $\Delta^*(f)$ -- set of points where $f$ has a finite two-sided derivative -- and $\Delta(f)$ -- set of points where $f$ has a finite or infinite two-sided derivative -- with subscripts of $+$ and $-$ for the corresponding right side and left side notions (e.g. p. 69 of Garg's book Theory of Differentiation). I've also seen $D(f)$ and $\mathcal D (f)$ for these sets of points. However, these uses are in advanced and specialized literature, and probably not in the literature you're referring to, otherwise you probably would have encountered some of the notation I've mentioned.

Incidentally, $D(f)$ and $\mathcal D(f)$ are also sometimes used for the set of discontinuity points of a function, as well as for the set of Darboux points of a function, and this is probably why some authors use the $\Delta$ symbol.

To further muddy the situation, $\Delta'$ is used for the set of functions having a finite and two-sided derivative at every point (e.g. p. 14 of Bruckner's book Differentiation of Real Functions), and I've also seen $D$ and $\mathcal D$ used for these sets of functions.

My advice is to NOT introduce notation for such sets of points unless the set of points plays an essential role in what you are writing about (such as I did in this MSE answer), since introducing such notation when most other writers don't will make your writing more opaque to others trying to quickly scan over the paper/book for some result. It also increases the symbolic clutter of the text, something I've noticed many here seem to think is a sign of increased rigor -- it isn't, at least not if you pay attention to words used and say what you mean and not mean what you say. For an example of what I mean about symbolic clutter and readability and (logical) rigor, see my answer to How formal or informal should math texts (written for different purposes) be?

Dave L. Renfro
  • 36,843