Continuous function, function y = f (x) When the change of independent variable x is small, the change of dependent variable y is also small. For example, the temperature changes with time, as long as the time change is small, the temperature change is small; For another example, the displacement of a free-falling body changes with time, and as long as the time change is short enough, the displacement change is very small. For this phenomenon, we say that the dependent variable changes continuously relative to the independent variable, which can be strictly described by the limit: let the function y = f (x) be defined near the x0 point, and if so, let the function f be continuous at the x0 point. If the function defined in the interval I is continuous at every point x∈I, then F is said to be continuous at I, at this time, it looks like a continuous curve in the rectangular coordinate system without fracture.

In the development history of mathematical analysis, mathematicians have always speculated that a continuous function is derivable within its defined interval, except for a few points at most. In other words, the nondifferentiable points of a continuous function are at most countable sets.

At that time, due to the limited means of expression of functions, this conjecture was correct only from the perspective of elementary functions or piecewise elementary functions. However, with the development of series theory, the means of function expression has expanded, and mathematicians can express a wider range of functions through function term series. Wilstrass is a master of series theory. In 1872, he first constructed a continuous and non-derivative function by using the series of function terms, which made a negative end to the above speculation (see figure for the formula).

After the construction of Weierstrass counterexample, it caused a great shock in the field of mathematics, because the traditional mathematical methods were powerless to this function, which made classical mathematics fall into another crisis. But in turn, the crisis has prompted mathematicians to think about new methods to study this kind of functions, thus contributing to the emergence of a new discipline "fractal geometry". The so-called "fractal" refers to a "shape" in geometry, and its part is similar to the whole to some extent. This property of "shape" is also called "self-similarity"

We know that the object of classical geometry research is regular and smooth geometric figures, but there are many irregular and unsmooth geometric figures in nature, all of which have the above-mentioned "self-similarity". Like a cloud boundary; The outline of the mountain peak; A grotesque coastline; A winding river; Irregular cracks in materials and so on. Although these endless curves are continuous everywhere, they are not necessarily derivable everywhere. Therefore, "fractal geometry" has attracted the attention of mathematicians since its birth, and soon developed into a new discipline with broad application prospects.

Wilstras

Karl Theodore William Karl Theodor Wilhelm Weierstra? , whose surname can be written as Weierstrass,181510/0/0/3/012/897 19), a German mathematician, is known as "modern". Hostains Feld (now German) was born in westfalen and died in Berlin.

Karl Veiershtrass's father is a government official Wilhem Weierstrass and his mother is Tiodola von der Forster. He became interested in mathematics when he was studying in middle school, but after graduating from high school, he entered Bonn University to work in a government department. What he wants to study is law, economy and finance, which goes against his desire to study mathematics. His solution to the contradiction is not to pay attention to the assigned courses, but to continue to teach himself mathematics in private. As a result, he left the university without getting a degree. His father found a place for him in a teacher training school in Mü nster, and he later registered as a teacher in the city. During his study, he took Christoph Gudermann's class and became interested in elliptic functions.

After 1850, Veiershtrass was ill for a long time, but he still published a paper, which made him famous. 1857 Berlin University gave him a math chair.

1854 published a monograph on the development of Abelian function theory-On Abelian function theory, which was made public. According to his academic achievements, Konigsberg University awarded him an honorary doctorate. 1856 was recommended by Cuomo as the University of Berlin (Freie Universit? T Berlin) assistant professor, 1865 promoted to professor. Before his death, most of his research results were taught and disseminated to students. 1886 published a collection of essays on function theory. Although his works are not many, he has published the most influential papers.

Wilstrass's main contributions are in mathematical analysis, analytic function theory, variational method, differential geometry and linear algebra. He is a master who introduces strict arguments into analysis. His critical spirit had a great influence on mathematics in19th century. Based on strict logic, he established the theory of real numbers, defined irrational numbers with monotone bounded sequences, and gave strict definitions of upper and lower bounds, limit points and continuous functions of number sets. In 186 1, he also constructed a famous continuous function with differentiability everywhere, which made an important contribution to the arithmeticization of analysis. He completed the definition of limit described by inequality introduced by Cauchy (so-called ε-δ definition). In analytic function theory, Wilstrass also made important contributions. He established the power series expansion theorem of analytic function and the basic theory of multivariate analytic function, and made some achievements in algebraic function theory and Abel integral. In the variational method, he gave the variational structure with parametric function and studied the discontinuous solution of the variational problem. In differential geometry, he studied geodesics and minimal surfaces. In linear algebra, elementary factor theory is established and used to simplify matrices. He is also an outstanding educator, who has trained a large number of outstanding mathematical talents in his life, including Kovalevskaya, Schwartz, Tammy-Levreux, Schottky and fuchs.