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December 5

Non-calculic functions

Just as algebraic numbers are numbers related to the integers by a finite number of additions, subtractions, multiplications, divisions, and taking nth roots where n is a positive integer; calculic functions can be defined as functions related to the polynomials by a finite number of additions, subtractions, multiplications, divisions, compositions, taking the inverse, differentiations, and integrations. Are there any functions that are not calculic that have a special name?? Or is there a proof that all mathematical functions are calculic?? Georgia guy (talk) 02:22, 5 December 2020 (UTC)[reply]

Not all functions can be generated in this way. Any such function will be almost-everywhere continuous, so "most" functions (in a number of different senses) will not be of this form. Even amongst continuous functions, your calculic functions form a separable subspace (the functions generated from starting with rational polynomials form a dense subset), so can't be everything.--2406:E003:E0A:7A01:4C9:12F8:6BC4:9301 (talk) 03:13, 5 December 2020 (UTC)[reply]

It occurs to me that you need to be careful with which convergence notion you use to say that they're separable. Still, one can make a continuous function that grows faster than all your calculic functions.--2406:E003:E0A:7A01:4C9:12F8:6BC4:9301 (talk) 04:47, 5 December 2020 (UTC)[reply]

The Weierstrass function is one such function which (obviously) has a special name. Also almost any function based on fractals will fit the bill, for example Minkowski's question-mark function. --RDBury (talk) 05:05, 5 December 2020 (UTC)[reply]

PS. I'm not sure if this counts or not, but some differential equations can't be solved using ordinary methods, so the corresponding solutions might be included as well. In particular when chaotic phenomena are involved, such as with the Lorenz system. Also, I'm guessing the potential function for the Mandelbrot set would qualify as well. I'm also thinking the Riemann zeta function would qualify. --RDBury (talk) 05:28, 5 December 2020 (UTC)[reply]

The concept of "calculic function" seems to be close to that of "Liouvillian function", but I'm not sure they coincide. Is the inverse error function Liouvillian? Next to the functions that have already been mentioned, the sign function is a named well-known function that is non-calculic, as are the floor and ceiling functions. These are defined everywhere yet have jump discontinuities. Nowhere differentiable functions can also not be calculic; the Weierstrass function and Minkowski's question-mark function fall into this class. The Cantor function is almost the opposite: it is differentiable almost everywhere and then its derivative vanishes, yet it is non-constant: it is a singular function. The Dirichlet function is even nowhere continuous. Thomae's function is somewhat similar; although continuous almost everywhere, its discontinuities form a dense set. Calculic functions cannot have any such weirdnesses. --Lambiam 08:04, 5 December 2020 (UTC)[reply]

Not that this is central to the discussion, but the first sentence contains a false premise ("algebraic numbers are related ..."); for example, the unique real root of $x^{5}+x+1=0$ is algebraic but is not related to the integers by the operations described. (The error is that "taking roots" needs to be replaced with the more general ability to solve polynomial equations.) --JBL (talk) 16:28, 5 December 2020 (UTC)[reply]