A: History shows that important mathematical concepts have an inestimable effect on the development of mathematics, and the influence of function concepts on the development of mathematics can be said to be lasting and extraordinary. Looking back at the historical development of the concept of function and seeing the historical process in which the concept of function has been continuously refined, deepened and enriched is a very beneficial thing, which not only helps us to improve the clarity of our understanding of the context of the concept of function, but also helps us to understand the great role of mathematical concepts in mathematical development and learning. (1) Marx once thought that the concept of function originated from the study of indefinite equations in algebra. Since Diophantine in Roman times had studied indefinite equations, the concept of function had sprouted at least then. Since Copernicus's astronomical revolution, sports has become a common problem for Renaissance scientists. People are thinking: since the earth is not the center of the universe, it has its own. The orbit of this planet is elliptical. What is the principle? In addition, it is not only a problem that scientists try to solve, but also a problem that military strategists ask to solve to study the route, range and height that projectiles can reach on the surface of the earth, as well as the influence of projectile speed on height and range. The concept of function is a mathematical concept derived from the study of motion, and motion is the mechanical source of the concept of function. (2) Mathematicians had contacted and studied many specific functions long before the concept of function was clearly put forward. Such as logarithmic function, trigonometric function, hyperbolic function, etc. Descartes had noticed the dependence of one variable on another in analytic geometry around 1673, but at that time he didn't realize the need to refine the general concept of function, so mathematicians didn't know the general meaning of function until Newton and Leibniz established calculus in the late17th century. 10016.0000000000606 Leibniz first used the word "function" to represent "power", and later he used it to represent the geometric quantities of various points in the world. It can be seen that the original mathematical meaning of the word "function" is quite extensive and vague. Almost at the same time, Newton used another term "flow" to express the relationship between variables in the discussion of calculus until 1689. Swiss mathematician johann bernoulli clearly defined the concept of function on the basis of Leibniz's concept of function. Bernoulli called the quantity formed by variable x and constant in any way "the function of x" and expressed it as yx. At that time, because the operations of connecting variables and constants were mainly arithmetic operations, trigonometric operations, exponential operations and logarithmic operations, Euler simply named the formula formed by connecting variables X and constants C with these operations as analytic functions. It is divided into "algebraic function" and "transcendental function". /kloc-in the middle of the 8th century, D'Alembert and Euler introduced the concept of "arbitrary function" successively because of the study of string vibration. When D'Alembert explained the concept of "arbitrary function", he said that it meant "arbitrary analytical formula", while Euler thought it was "arbitrarily drawn curve". Now it seems that these are all functions. It is an extension of the concept of function. (3) The concept of function lacks scientific definition, which leads to sharp contradiction between theory and practice. For example, partial differential equations are widely used in engineering technology, but the lack of a scientific function definition greatly limits the establishment of partial differential equation theory. From 1943 to 1983, Gauss began to turn his attention to physics. He invented the telegraph with Wilbur. I have done a lot of magnetic experiments and put forward the important theory that "force is inversely proportional to the square of distance", which makes the function appear as an independent branch of mathematics. The actual need urges people to further study the definition of function. Later, people gave the definition that if one quantity depends on another quantity, the former quantity will change with the latter quantity. Then the first quantity is called a function of the second quantity. "Although this definition has not revealed the essence of function, it is gratifying progress to inject change and movement into the definition of function." In the history of the development of function concept, the work of French mathematician Fourier has the greatest influence. Fourier profoundly reveals the essence of function and thinks that function need not be limited to analytical expressions. In his representative work, Analytical Theory of Heat, he wrote. 1822. They are close in every way. In this book, he expressed the function given by a discontinuous "line" in the form of the sum of trigonometric series. More precisely, any function with a period of 2π can be represented by [-π, π] interval, in which the research of Fourriere fundamentally shook the old traditional concept of function and caused it in the field of mathematics at that time. There is no insurmountable gap between analytical formula and curve. The viewpoint that series connects analytical formula and curve and function is analytical formula eventually becomes a huge obstacle to reveal the relationship between functions. Through an argument, the function definitions of Lobachevsky and Dirichlet came into being. 19438+0834, Russian mathematician Lobachevsky put forward the definition of function: "The function of x is such a number, which has a certain value for each x, and it changes with x, and the function value can be given by an analytical formula or a condition, which provides a method to find all the corresponding values. This dependence of the function can exist, but it is still unknown. "This definition establishes the correspondence between variables and functions, which is a significant development of the concept of functions, because" correspondence "is the essential attribute and core part of the concept of functions. According to this definition, even if it is expressed as follows, it is still said to be a function (Dirichlet function): f(x)= 1 (x is a rational number), 0 (x is an irrational number). Dirichlet's definition of function avoids all descriptions of dependence in previous function definitions and is unconditionally accepted by all mathematicians in a completely clear way. At this point, we can say the concept of function and the essential definition of function. This is the definition of classical function that people often say. (4) The further development of production practice and scientific experiment has caused new sharp contradictions in the concept of function. In the 1920s, people began to study microphysical phenomena. 1930 quantum mechanics came out, and a new function-δ function was needed in quantum mechanics, that is, ρ (x) = 0, x≠0. It caused a heated debate among people. According to the original definition of function, only the correspondence between numbers is allowed, and "∞" is not regarded as a number. In addition, it is incredible that the independent variable has only one non-zero function, but its integer value is not equal to zero. However, the δ function is really an abstraction of the actual model. For example, when cars and trains pass through the bridge, they naturally put pressure on the bridge. Theoretically, there is only one contact point between the wheels of a vehicle and the bridge deck. Let the pressure of vehicles on the track and bridge deck be a unit. At this time, the pressure at the contact point x=0 is P(0)= pressure/contact surface = 1/0 =∞. At the rest point x≠0, there is no pressure because there is no pressure, that is, P(x)=0. In addition, there is a new definition of modern function: if any element X of set M always has an element Y determined by set N corresponding to it, it is said that a function is defined on set M, which is denoted as y=f(x). Element x is called independent variable and element y is called dependent variable. Although the modern definition of function is only a few words apart from the classical definition in form, it is a major development in concept and a major turning point in the development of mathematics. Modern functional analysis can be used as a sign of this turning point. It studies the functional relations on general sets. The definition of function has been tempered and changed for more than 200 years, forming a modern definition of function, which should be said to be quite perfect. However, the development of mathematics is endless, and the formation of modern definition of function does not mean the historical end of the development of function concept. In the past twenty years, mathematicians have simplified functions into a broader concept-"relationship". We define the product set of x and y as x x y = {(x, y) | x ∈ x, y}. A subset of the product set x x y is called a relationship between x and y. If (x, y) | x ∈ y} If (x, y), (x, z)∈f must have y=z, then f is called a function from x to y. In this definition, "correspondence" has been avoided formally. From the whole process of the development of the above-mentioned function concept, we realized that we should study and study with practice and a lot of mathematical data.