Function of a real variable

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers

Most real functions that are considered and studied are differentiable in some interval.

That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an

The image of a function of a real variable is a curve in the codomain.

It is generally assumed that the domain contains an interval of positive length.

This is the case of: A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x).

To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.

, the domain of the function, which is always supposed to contain an interval of positive length.

is the set of all values of f when the variable x runs in the whole domain of f. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value.

The preimage of a given real number y is the set of the solutions of the equation y = f(x).

This means that it is not worthy to explicitly define the domain of a function of a real variable.

which is a function only if the set of the points (x) in the domain of f such that f(x) ≠ 0 contains an open subset of

Until the second part of 19th century, only continuous functions were considered by mathematicians.

At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces.

As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.

[1] Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that for all x in the domain such that If the limit exists, it is unique.

If a is in the interior of the domain, the limit exists if and only if the function is continuous at a.

A real-valued implicit function of a real variable is not written in the form "y = f(x)".

At a point r(t = c) = a = (a1, a2, ..., an) for some constant t = c, the equations of the one-dimensional tangent line to the curve at that point are given in terms of the ordinary derivatives of r1(t), r2(t), ..., rn(t), and r with respect to t: The equation of the n-dimensional hyperplane normal to the tangent line at r = a is: or in terms of the dot product: where p = (p1, p2, ..., pn) are points in the plane, not on the space curve.

The physical and geometric interpretation of dr(t)/dt is the "velocity" of a point-like particle moving along the path r(t), treating r as the spatial position vector coordinates parametrized by time t, and is a vector tangent to the space curve for all t in the instantaneous direction of motion.

Similarly, in special relativity, the Lorentz transformation matrix for a pure boost (without rotations): is a function of the boost parameter β = v/c, in which v is the relative velocity between the frames of reference (a continuous variable), and c is the speed of light, a constant.

In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply.

This occurs especially often in quantum mechanics, where one takes the derivative of a ket or an operator.

This occurs, for instance, in the general time-dependent Schrödinger equation: where one takes the derivative of a wave function, which can be an element of several different Hilbert spaces.

A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values.

In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.

The cardinality of the set of real-valued functions of a real variable,

, which is strictly larger than the cardinality of the continuum (i.e., set of all real numbers).

This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.

[2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable.