Numerical function

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

The theorems of real analysis rely intimately upon the structure of the real number line. The real number system consists of a set (${\displaystyle \mathbb {R} }$), together with two operations (+ and •) and an order (<), and is, formally speaking, an ordered quadruple consisting of these objects: ${\displaystyle (\mathbb {R} ,+,\bullet ,<)}$. There are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. In particular, the property of completeness distinguishes the real numbers from other ordered fields (e.g., the rational numbers ${\displaystyle \mathbb {Q} }$) and is critical to the proof of several key properties of real-valued functions. The completeness of the reals is often conveniently expressed as the least upper bound property (vide infra). Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in more detail in the main article.

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