Dirichlet function On Baidu Encyclopedia The Dirichlet function over the real number field is expressed as: D(x)=lim(n→∞){lim(m→∞)[cosπm!x]^n} (1) It can also simply express the form of piecewise function d (x) = 0 (x is irrational) or 1 (x is rational) (2) The process of finding (1) and pushing (2)
A little difficult
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- 1. Some problems on Dirichlet function For: 1. Is a periodic function, 3 is a period of it; 2. The equation f (x) = cosx has rational roots; 3. The equation f [f (x)] = f (x) has the same solution set as the equation f (x) = 1 In fact, Dirichlet function refers to: (1) when x is a rational number, f (x) = 1; (2) when x is an irrational number, f (x) = 0
- 2. Why can't Dirichlet higher numbers be expressed as limit functions of continuous functions
- 3. Why is Dirichlet function not continuous? It is said that Dirichlet function is discontinuous everywhere According to the definition of continuity, if f (x0) = LIM (x - > x0) f (x), the function is continuous at x0 For example, it is known that x0 belongs to Q. if it is not continuous, LIM (x - > x0) must not belong to Q. how to verify that LIM (x - > x0) does not belong to q?
- 4. Is Dirichlet function continuous almost everywhere on R? I know it's discontinuous everywhere. Is it continuous almost everywhere in real change?
- 5. Using the mixed operation of addition and subtraction of rational numbers, (- one fifth) + two fifths + (- three fifths) Using the method of mixed operation of addition and subtraction of rational numbers, 1. (- one fifth) + two fifths + (- three fifths) 2.(-7)-(-5)+(-4)-(-10) 3.4.7-(-8.9)-7.5+(-6) 4. - half + (- one sixth) - (- one fourth) - (+ two thirds) 5. - half - five and one fifth + 4.5 + Half - 4.5 + five and one fifth 6.(-2.5)-(+2.7)-(-1.6)-(-2.7)+(+2.4) 7. 3 / 4-7 / 2 + (- 1 / 6) - (- 2 / 3) - 1
- 6. If the sum of products of two numbers is known and one of them is - 2 and 3 / 7, find another number Given that the quotient of two numbers is - 3 and 1 / 2, and one of them is 2 and 1 / 3, find another number The title says that the product of two known numbers is 1, so it's wrong.
- 7. Write a number whose product with 2 √ 3 is a rational number
- 8. There are only four rational numbers, of which the sum of every three numbers is 3, 5, 13 and 15 respectively. How much is the product of these four rational numbers emergency
- 9. 1. If the rational numbers a and B are opposite to each other, then the sum of the two numbers is? If the rational numbers a and B are reciprocal to each other, then the product of the two numbers is? 2. Calculation (- 0.25) x0.5x (- seventy and three fifths) X4 (- one and two thirds) x (- 18) - 15x one and two thirds Ten and four fifths x (- 25) We need the formula!
- 10. The following statement is wrong () A. Any rational number has reciprocal B. the product of reciprocal numbers is 1C. Reciprocal numbers have the same sign D. 1 and - 1 are reciprocal numbers
- 11. The famous Dirichlet function is defined like this What are the independent variables and the dependent variables of this function What are the definition and value ranges of this function Please write the function values when x = - 1, root 2, 6.4 and 3.1415 respectively
- 12. Besides Dirichlet functions, which functions are Riemann non integrable and Lebesgue integrable,
- 13. How to prove that Dirichlet function has no limit?
- 14. What are the common forms of periodic functions Excuse me?
- 15. Several expressions of periodic function? For example, f (x + 2) = f (x) f (X-2) = f (x) f (x + 2) = - f (x) f (X-2) = - f (x)... And so on! Are they periodic functions? If so, what are the axis of symmetry and the period? What if they are even or odd functions? What are the axis of symmetry and the period? What are the expressions of periodic functions? Detailed answer to chase points!
- 16. What is the fundamental period of a function
- 17. On the small problem of function period If f (x-4) = - f (x), f (x) is an odd function, why can we get f (X-8) = f (x) -F (x-4) = f (X-8) why
- 18. How to find the minimum value and the minimum positive period of a function
- 19. How to find the minimum positive period in positive metaphysical function?
- 20. Let f (x) = cos (2 π − x) + 3cos (π 2 − x), then the minimum positive period of the function is () A. π2B. πC. 2πD. 4π