# A report onCalculus, Mathematical analysis and Mathematics

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

- Calculus

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

- Mathematical analysis

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.

- Mathematical analysis

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

- Calculus

## Function (mathematics)

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).

Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain.

## Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value.

Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion).

The development of calculus in the 18th century used the entire set of real numbers without having defined them rigorously.

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built.

## Geometry

Geometry is, with arithmetic, one of the oldest branches of mathematics.

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.