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 ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval.The most widely considered such functions are the real functions, 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. c a r d ( R R ) = c a r d ( R ) c a r d ( R ) = c c = ( 2 ℵ 0 ) c = 2 ℵ 0 ⋅ c = 2 c {displaystyle mathrm {card} (mathbb {R} ^{mathbb {R} })=mathrm {card} (mathbb {R} )^{mathrm {card} (mathbb {R} )}={mathfrak {c}}^{mathfrak {c}}=(2^{aleph _{0}})^{mathfrak {c}}=2^{aleph _{0}cdot {mathfrak {c}}}=2^{mathfrak {c}}} . 2 c = c a r d ( 2 R ) ≤ c a r d ( X R ) ≤ c a r d ( R R ) = 2 c {displaystyle 2^{mathfrak {c}}=mathrm {card} (2^{mathbb {R} })leq mathrm {card} (X^{mathbb {R} })leq mathrm {card} (mathbb {R} ^{mathbb {R} })=2^{mathfrak {c}}} . c a r d ( C 0 ( R ) ) ≤ c a r d ( R Q ) = ( 2 ℵ 0 ) ℵ 0 = 2 ℵ 0 ⋅ ℵ 0 = 2 ℵ 0 = c {displaystyle mathrm {card} (C^{0}(mathbb {R} ))leq mathrm {card} (mathbb {R} ^{mathbb {Q} })=(2^{aleph _{0}})^{aleph _{0}}=2^{aleph _{0}cdot aleph _{0}}=2^{aleph _{0}}={mathfrak {c}}} . 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 ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval.The most widely considered such functions are the real functions, 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, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of ℝ-vector space over the reals. 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 ℝ-algebra, such as the complex numbers or the quaternions. The structure ℝ-vector space of the codomain induces a structure of ℝ-vector space on the functions. If the codomain has a structure of ℝ-algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications. A real function is a function from a subset of R {displaystyle mathbb {R} } to R , {displaystyle mathbb {R} ,} where R {displaystyle mathbb {R} } denotes as usual the set of real numbers. That is, the domain of a real function is a subset R {displaystyle mathbb {R} } , and its codomain is R . {displaystyle mathbb {R} .} It is generally assumed that the domain contains an interval of positive length. For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous and differentiable at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of: