Function is one of the basic concepts of modern mathematics and the main research object of higher mathematics. In the real world, everything moves in a certain space. In the 17th century, mathematics first introduced the basic concept of function from the study of motion (such as astronomy, navigation, etc.). For more than 200 years since then, this concept has occupied a central position in almost all scientific research.
In the 17th century,Galileo Galilei almost entirely included the concept of function or variable relationship in his book "Two New Sciences", expressing the relationship of functions in the language of words and proportions. Around 1637,Descartes had noticed the dependence of one variable on another variable in his analytic geometry, but he had not realized the need to refine the concept of functions at that time. Therefore, it was not until the late 17th century thatNewton andLeibniz established micro At the time of integration, no one had yet understood the general meaning of functions, and most functions were studied as curves. In 1673, Leibniz first used " Function " to represent " power " .Later, he used the word to . At the same time, Newton used represent" flow " to express the relationship between variables in his discussions of calculus .
In the 18th century, in 1718,Johann Bernoulli further defined the concept of function based on Leibniz's concept of function: " A quantity composed of any variable and any form of a constant. " What he meant Any formula composed of a variable and a constant is called a function of , and it is emphasized that the function must be expressed by a formula. In 1748, Euler defined a function in his book "Introduction to Infinite Analysis" as: " The function of a variable is an analytic expression composed of some numbers or constants of the variable and any method. " He referred to John The function definition given by Bernoulli is called an analytic function, and he further distinguishes it into algebraic functions and transcendental functions, and also considers " arbitrary functions " . It is not difficult to see that the definition of function given by Euler is more general and has wider significance than the definition of Johann Bernoulli. In 1755, Euler gave another definition: " If some variables depend on other variables in a certain way, that is, when the latter variables change, the previous variables also change, we call the previous variables is a function of the following variables. "
In the 19th century, in 1821,Cauchy gave the definition of a function starting from the definition of variables: " There is a certain relationship between certain variables. Once the value of one of the variables is given, the values of other variables can follow. When determined, the initial variable is called an independent variable, and the other variables are called functions. " In Cauchy's definition, the word independent variable first appeared, and he also pointed out that there is no need for an analytical expression for a function. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a big limitation. In 1822,Fourier discovered that certain functions can be represented by curves, one formula, or multiple formulas, thus ending the debate on whether the concept of a function can be represented by a single formula and bringing the understanding of functions to the forefront. It has advanced to a new level. In 1837 , Dirichlet broke through this limitation and believed that it does not matter how to establish the relationship between and . He expanded the concept of function and pointed out: " For every certain value of in a certain interval , is has a definite value, then, is called a function of . "the function definition that we learn in advanced mathematics or calculus courses .
