The Early Concept of 1. Function —— Function under the Concept of Geometry

/kloc-Galileo in the 0/7th century (Italy, 1564- 1642), in his book "Two New Sciences", almost all contained the concept of the relationship between functions or variables, and expressed the relationship between functions in the language of words and proportions. Descartes (France, 1596- 1650) noticed the dependence of one variable on another around his analytic geometry 1673. However, because he didn't realize that the concept of function needed to be refined at that time, no one had defined the function until Newton and Leibniz established calculus in the late17th century.

1673, Leibniz first used "function" to express "power". Later, he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate, tangent length and so on. At the same time, Newton used "flow" to express the relationship between variables in the discussion of calculus.

2.18th century function concept-function under algebraic concept.

17 18 johann bernoulli (Swiss, 1667- 1748) defined the concept of function on the basis of the concept of Leibniz function: "a quantity consisting of any variable and any form of constant." He means that any formula composed of variable X and constant is called a function of X, and he emphasizes that functions should be expressed by formulas.

1755, Euler (L. Euler, Switzerland, 1707- 1783) defines a function as "if some variables depend on other variables in some way, that is, when the latter variable changes, the former variable also changes, and we call the former variable a function of the latter variable."

Euler (L. Euler, Switzerland, 1707- 1783) gave a definition: "The function of a variable is an analytical expression composed of this variable and some numbers or constants in any way." He called the function definition given by johann bernoulli's analytic function, and further divided it into algebraic function and transcendental function, which was also considered as "arbitrary function". It is not difficult to see that Euler's definition of function is more universal and extensive than johann bernoulli's.

3./kloc-the concept of function in the 0/9th century-the function under correspondence.

182 1 year, Cauchy (France, 1789- 1857) gave a definition from the definition of variables: "Some variables have certain relationships. When the value of one variable is given, the values of other variables can be determined accordingly, so the initial variable is called independent variable. The word independent variable appeared for the first time in Cauchy's definition, and pointed out that functions don't need analytic expressions. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a great limitation.

1822, Fourier (France,1768-1830) found that some functions have also been expressed by curves, or they can be expressed by one formula, or they can be expressed by multiple formulas, thus ending the debate on whether the concept of functions is expressed by only one formula and pushing the understanding of functions to a new level.

1837, Dirichlet (Germany, 1805- 1859) broke through this restriction and thought that it was irrelevant how to establish the relationship between x and y. He broadened the concept of function and pointed out: "For every definite value of X in a certain interval, Y has a definite value, so this definition of Y avoids the description of dependency in the definition of function and is accepted by all mathematicians in a clear way. This is what people often call the classic function definition.

After the set theory founded by Cantor (German, 1845- 19 18) played an important role in mathematics, veblen (American, veblen, 1880- 1960) used "set" and ".

4. Modern function concept-function under set theory.

F. Hausdorff defined the function in 19 14 with the fuzzy concept of "ordered couple" in the outline of set theory, avoiding the two fuzzy concepts of "variable" and "correspondence". In 192 1, Kuratowski defined "ordered pair" with the concept of set, which made Hausdorff's definition very strict.

In 1930, the new modern function is defined as "If there is always an element Y determined by set N corresponding to any element X of set M, then a function is defined on set M, and it is denoted as y=f(x). Element x is called an independent variable and element y is called a dependent variable. "