a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous...

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as...

mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies...

if f {\displaystyle f} is Cauchy-continuous. It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to X {\displaystyle...

the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is Cartesian...

holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral...

derivatives of u and v satisfy the Cauchy–Riemann equations at that point. A holomorphic function is a complex function that is differentiable at every point...

arguments were introduced into calculus. Here Cauchy defined continuity as follows: The function f(x) is continuous with respect to x between the given limits...

Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function ⁠ f {\displaystyle f} ⁠, this is equivalent to...

In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would...

mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively...

mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat)...

it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent...

an additive function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is linear if: f {\displaystyle f} is continuous (Cauchy, 1821). In...

approximations are cumulative distribution functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively. Approximations...

Let f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } be a continuous function on the closed interval [ a , b ] {\displaystyle [a,b]} , and differentiable...

result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution...

mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain...

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between...

functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the...

Augustin-Louis Cauchy include: Bolzano–Cauchy theorem Cauchy boundary condition Cauchy completion Cauchy-continuous function Cauchy–Davenport theorem Cauchy distribution...

analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either...

which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes...

{\displaystyle \xi } produces the frequency-domain function, and it converges at all frequencies to a continuous function tending to zero at infinity. If f ( x )...

a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given...

locally the graph of a function. Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini...

known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions...

distribution function or c.d.f.) or inverse distribution function. With reference to a continuous and strictly increasing cumulative distribution function (c.d...

1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is...

characteristic function of a continuous random variable X {\displaystyle X} is the Fourier transform of its probability density function f X ( x ) {\displaystyle...