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A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite .
This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than (hence a set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.
The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function. [7] In the case , the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle. The continued fractions on the right hand side will ...
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
Every Lipschitz continuous map is uniformly continuous, and hence continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if { f n } is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
This is the case, for example, for continuous functions on a topological space; for k-times differentiable, smooth, or analytic functions on a real manifold (when such functions are defined); for holomorphic functions on a complex manifold; and for regular functions on an algebraic variety.
A bump function is a smooth function with compact support.. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.
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