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Every rational function can be naturally extended to a function whose domain and range are the whole Riemann sphere (complex projective line). Rational functions are representative examples of meromorphic functions. Iteration of rational functions (maps) [5] on the Riemann sphere creates discrete dynamical systems.
A function f from X to Y. The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
Holomorphic function: complex-valued function of a complex variable which is differentiable at every point in its domain. Meromorphic function: complex-valued function that is holomorphic everywhere, apart from at isolated points where there are poles. Entire function: A holomorphic function whose domain is the entire complex plane ...
Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function , whose graph is a hyperbola, and whose domain is the whole real line except for 0.
Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
Given a holomorphic function f on the blue compact set and a point in each of the holes, one can approximate f as well as desired by rational functions having poles only at those three points. In complex analysis , Runge's theorem (also known as Runge's approximation theorem ) is named after the German mathematician Carl Runge who first proved ...
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