1 early function concept-function under geometric concept Galileo1Galileo in the 7th century (meaning 1564- 1642) almost always included the concept of the relationship between functions or variables, and expressed the relationship between functions in words and proportional languages. Descartes (France, 1596- 1650) noticed the dependence of one variable on another in his analytic geometry around 1673, but he didn't realize the need to refine the general concept of function at that time, so Newton and Leibniz didn't establish calculus until the late17th century.
2/kloc-function concept in the 8th century-function under algebraic concept johann bernoulli (Rui,1667-1748)1718 johann BernoulliJohann a clear definition of function concept on the basis of Leibniz function concept: from
Euler (L. Euler, Rui,1707-1783)1in the middle of the 8th century, Euler gave a very vivid function symbol, which has been used ever since. Euler's definition is that the function of a variable is an analytical expression composed of this variable and some numbers, that is, constants, in any way. He called johann bernoulli's definition of function analytic function, and further divided it into algebraic function (only referring to the algebraic operation between independent variables) and transcendental function (trigonometric function, logarithmic function and unreasonable power of variables), and also considered "arbitrary function" (representing the function of drawing any curve). It is not difficult to see that Euler's definition of function is more universal and broader than johann bernoulli's definition. /kloc-the concept of function in the 10th/9th century-the function under the corresponding relation (Fourier, method,1768-1830)1822 Fourier found that some functions can be expressed by curves, one formula or multiple formulas, thus ending the debate on whether the concept of function can be expressed by a single formula. In 1823, Cauchy (method, 1789- 1857) gave the definition of function from the definition of variable, and pointed out that although infinite series is an effective method to specify a function, it is not necessary for a function to have analytical expressions, but he still thinks that the function relationship can be expressed by multiple analytical expressions, which is a big one.
Dirichlet (Germany,1805-1859)1837 Dirichlet thinks that how to establish the relationship between x and y is irrelevant. He expanded the concept of function and pointed out: "For every definite value of X in a certain interval, Y has one or more definite values, so it is called X. Dirichlet's definition of function, which avoids all descriptions of dependence in previous function definitions, is concise and accurate, and is unconditionally accepted by all mathematicians in a completely clear way. At this point, it can be said that the concept of function and the essential definition of function have been formed, that is, the classical function definition that people often say.
Van Buren (USA, 1880- 1960) Van Buren gave the definition of modern function with the concepts of "set" and "correspondence". Through the concept of set, the correspondence, domain and range of functions are further concretized, breaking through the limitation that "variables are numbers". Variables can be numbers or other objects. 4 modern function concept-function under set theory F. F. Hausdorf) 19 14+04 In the Outline of Set Theory, F. Hausdorf defined the function with "ordered couple". Its advantage is that it avoids the fuzzy concepts of "variable" and "correspondence", and its disadvantage is that it introduces the fuzzy concept of "ordered pair". In 192 1, Kuratowski defined "ordered pairs" with the concept of sets, that is, ordered pairs (a, b) are sets {{a}, {b}}, thus making 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, it is said that a function is defined on set m, and it is denoted as y=f(x). Element x is called independent variable and element y is called dependent variable.
The definition of function has been tempered and changed for more than 300 years, forming a modern definition form of function, but this does not mean the historical end of the development of function concept. In the 1940s, for the need of physics research, a function called dirac-δ function was discovered. It is not zero at all, but its integral on the whole line is equal to 1, which is inconceivable under the original definitions of function and integral. However, because the concept of function is broad, with the development of other disciplines based on mathematics, the concept of function will continue to expand.