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...

of U. f is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function f {\textstyle f} . Generally...

Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists...

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

piecewise linear, piecewise smooth, piecewise continuous, and others are also very common. The meaning of a function being piecewise P {\displaystyle P} , for...

\mathbb {R} } , and upper semi-continuous if − f {\displaystyle -f} is lower semi-continuous. A function is continuous if and only if it is both upper...

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

or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are...

A Continuous Function Chart (CFC) is a graphic editor that can be used in conjunction with the STEP 7 software package or with other tools, such as CODESYS...

In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle...

⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere. A continuous function fails to be absolutely continuous if it fails to...

a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants...

analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological...

measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the ordinary limit with...

the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable...

a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the...

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in...

smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain. A function of class...

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

In mathematics, a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is symmetrically continuous at a point x if lim h → 0 f ( x + h )...

{\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 )...

sublinear function on X . {\displaystyle X.} Then the following are equivalent: p {\displaystyle p} is continuous; p {\displaystyle p} is continuous at 0;...

instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete...

functions and linear functions Sine and cosine functions Exponential function Some functions are defined everywhere, but not continuous at some points. For...

bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of...

is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the...

image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization...

the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a , b ] {\displaystyle...

bounded. For example, the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined above is a continuous function with compact support [...

, {\displaystyle x\in X\setminus A,} there exists a real-valued continuous function f : X → R {\displaystyle f:X\to \mathbb {R} } such that f ( x ) =...