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

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...

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

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...

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

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...

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...

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

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

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

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

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

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

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...

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

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...

intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a, b], then it takes on any given...

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

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

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...

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

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given...

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

}}k\neq 0,} where P V {\displaystyle PV} means taking the Cauchy principal value. The signum function can be generalized to complex numbers as: sgn ⁡ z = z...

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

translation and scale parameter of the wavelets vary continuously. The continuous wavelet transform of a function x ( t ) {\displaystyle x(t)} at a scale a ∈ R...

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...