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Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and integrability; not so much things like even/odd, one-to-one)? Something along the lines of what this diagram does for complex numbers.

Complex Number Venn Diagram

[The original of this (and more) can be found here.]

I would be grateful if you could direct me to any good resources that categorize types of functions in a systematic and succinct manner. Illuminating examples of the different types of functions (e.g. Weierstrass's continuous-everywhere-but-differentiable-nowhere function) and schematic clarity would be pluses.

Let me know if you need more information. Thanks!

Edit: I've look around more on this site at related questions (notably Are the smooth functions dense in either L^2 or L^1? and what is the cardinality of set of all smooth functions in $L^1$?) and found them intriguing and somewhat helpful. I could really use help putting all of these and many other pieces together, though. Any takers?

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Is there really a nice hierarchy? Sure you could split functions up into analytic ones, meromorphic ones, those functions with branch cuts... and then genuine monsters, but what's the point of a hierarchy? –  J. M. Aug 16 '11 at 4:37
I don't know if there's a hierarchy or interesting interactions of categories. I'm asking because I don't know this terrain very well and am hoping for a better and more unified understanding of the possibilities, intricacies, and surprises that exist. One thing I've thought of is how there's continuous>C^1>...>C^inf. I'm not sure what all else is going on. –  Justin Lanier Aug 16 '11 at 4:51
You might find this book somewhat helpful: books.google.com/… –  John M Aug 16 '11 at 10:09

2 Answers 2

up vote 1 down vote accepted

From the book A Second Course on Real Functions by van Rooij & Schikhof:

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The publisher allows you to read excerpts online: the introduction, where this figure is taken from, and the table of contents, including a four-page list of examples & counterexamples.

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This is really nice, Marius. These are interesting categories, and I like how the introduction gives examples of functions that lie in the different regions. Thanks for your answer! Also: are there any categories that you could think to add? –  Justin Lanier Sep 10 '14 at 17:19
You're welcome! And here is, incidentally, a similar diagram (Fig 3) on a completely different topic. –  Marius Kempe 2 days ago

In your diagram: delete the Imaginary axis, change Integer to "Polynomial with a finite number of terms", change Rational to Rational function, change Algebraic to Algebraic function. I'm not sure what Real corresponds to. On another axis you could have the C^1>C^2 ... as you mentioned. On another axis you could have the field of the polynomial, i.e. "polynomial over integers", "polynomial over real", "polynomial over complex", "polynomial over vector space", etc.

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