member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the...

which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless...

a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for...

may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering...

a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and...

second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix...

complex-valued function f on an arbitrary set X (is isomorphic to, and therefore, in that sense, it) can be considered as an ordered pair of two real-valued functions:...

and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity...

mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on...

absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. A real-valued function v on a...

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional...

mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle...

limit of af(x) as x approaches p is aL. If f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x)...

{\displaystyle X} , then the function is said to be bounded (from) below by B {\displaystyle B} . A real-valued function is bounded if and only if it...

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

intermediate value theorem is an existence theorem, based on the real number property of completeness, and states: If the real-valued function f is continuous...

In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member...

violation of the Baire category theorem. The continuity set of any real valued function is a Gδ subset of its domain (see the "Properties" section for a...

semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively...

scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with...

function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear function is a function defined...

mathematics, a constant function is a function whose (output) value is the same for every input value. As a real-valued function of a real-valued argument, a constant...

transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach...

f(x_{2}))}{d_{X}(x_{1},x_{2})}}\leq K.} For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant...

This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized...

related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects...

In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function domain of elements that are not mapped to zero...

{\displaystyle f\colon {\mathcal {D}}\to \mathbb {R} } be a real-valued function. The sensitivity of a function, denoted Δ f {\displaystyle \Delta f} , is defined...

complex-valued function that has no negative frequency components.  The real and imaginary parts of an analytic signal are real-valued functions related...

variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable f ( x )...