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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). A complex rational function with degree one is a Möbius transformation. Rational functions are representative examples of meromorphic functions. [3]
A function is continuous if it is continuous at every point of its domain. The limit of a real-valued function of a real variable is as follows. [1] Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted = (),
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
For example, a quadratic for the numerator and a cubic for the denominator is identified as a quadratic/cubic rational function. The rational function model is a generalization of the polynomial model: rational function models contain polynomial models as a subset (i.e., the case when the denominator is a constant).
Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration. [ 4 ] Since every rational number has a unique representation with coprime (also termed relatively prime) p ∈ Z {\displaystyle p\in \mathbb {Z} } and q ∈ N {\displaystyle q\in \mathbb {N ...
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.
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.
Since rational functions with no poles are simply polynomials, we get the following corollary: If K is a compact subset of C such that C\K is a connected set, and f is a holomorphic function on an open set containing K, then there exists a sequence of polynomials () that approaches f uniformly on K (the assumptions can be relaxed, see Mergelyan ...