User:CRGreathouse/Functions
| Type | Properties | Closure | Examples | Limitations |
|---|---|---|---|---|
| Polynomial function | Addition, subtraction, multiplication, composition | x^2 - 3x + 1 | ||
| Rational function | field operations, composition | (3x^2 + 1)/(2x - 1) | Division by zero | |
| Algebraic function | Meromorphic, decidable | field operations | sqrt(x), Bring radical | Abel–Ruffini theorem |
| High-school identities | 2^x | Tarski's high school algebra problem | ||
| Real + exponential | e^x - 1 | Tarski's exponential function problem | ||
| Elementary function | Continuous | Field operations, composition, derivative | |sin(sin(x))|, e^cosh(x) | Richardson's theorem |
| Liouvillian function | Field operations, composition, derivative, integral | erf(x), li(x) |