When I wrote my master's thesis, a professor who read it said that I should not use the phrase "A function of class $k$." but instead "A function of class $C^k$". I am not an expert about mathematical history of notations, but I read that in Geometric Measure Theory, H. Federer actually uses the first one, and it seems logical for me: I think that $C^k$ is the abbreviation for "of class $k$". Therefore, employing "class $C^k$" seems like a repetition. Or maybe the other notation is just not used any more and should simply be prohibited?

9$\begingroup$ A googlebooks search using the word "function" and the phrase "of class $k$" will show you "class $k$" is used in a variety of settings. Notation conventions tend to come and go, but I'm willing to bet that "of class $k$" will be a lot less meaningful 50 years from now than "of class $C^{k}$". $\endgroup$– Dave L RenfroSep 15, 2015 at 18:25

30$\begingroup$ I always thought of the $C$ as standing for "continuouslydifferentiable" $\endgroup$– Eric WofseySep 15, 2015 at 18:25

13$\begingroup$ Federer could use a simplified notation in his book, if the term occurs very frequently. The standard notation is $C^k$. "Class $k$" will not be recognizable by most mathematicians. $\endgroup$– Alexandre EremenkoSep 15, 2015 at 18:37

$\begingroup$ Thank you Prof. Eremenko. The notation does not appear very often in Federer's book, but your second argument convinces me to use $C^k$. $\endgroup$– PaulBenjaminSep 15, 2015 at 18:47

10$\begingroup$ If "a function of class $C^k$" bothers you, you can always say "a $C^k$ function". $\endgroup$– Noah SteinSep 15, 2015 at 20:49
1 Answer
Federer was not exactly known, even to his contemporaries, for employing standard notation. Here is a quote from Steenrod's 1948 Math Review of some mimeographed notes of Federer for a course on differential geometry.
The most striking feature of the book to the casual reader is the notation. The author adopts the view that certain familiar notations are misleading, and obscure the meanings of definitions and theorems. He replaces them by more elaborate notations based on the roots in set theory of the concepts represented (e.g., the polynomial x becomes the sequence of its coefficients [0,1]). A few such changes would not be worthy of comment; but he has carried out the prodigious task of applying the same stern standards to every phase of the work. The result can be described by saying that a resemblance to any notation, living or dead, is purely coincidental.

29$\begingroup$ It's a guilty pleasure, how much I enjoy such barbed Math Reviews. $\endgroup$ Sep 15, 2015 at 19:45

2$\begingroup$ Thank you very much foliations, I did not know about this quote; it is really illuminating. As Federer worked on his PhD with the logician Anthony P. Morse, whose Wikipedia says that "His work is characterized by an unusual degree of formality.", maybe that explains why he is so formal. $\endgroup$ Sep 16, 2015 at 4:13

$\begingroup$ He's not the only one. Karl Menger invented a new notation and made consistent use of it and wrote much about it. Parts of it entered ordinary notation, parts didn't. Dirac vector covector notation is another example that however became popular, but whether a mathematician uses it depends on whether they feel like it, there are several standard options. Use the standard notation, unless your own notation is much better, in which case full speed ahead with your own and don't worry if somebody dislikes it. (If it's good others will like it too later.) Here AE is correct, $C^k$ is better. $\endgroup$ Sep 16, 2015 at 18:33

3$\begingroup$ One thing that might not be obvious from the quote given above is that most of the review is positive ("The author has performed a sizable task by presenting, for the first time, an organized account of material so widely distributed in the literature" and "A notable feature of the book is the use of invariant methods throughout"). $\endgroup$ Sep 16, 2015 at 22:04