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?
[Math] Mathematical writing : using an “out-of-date” notation
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Best 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.