Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.