How to prove that Dirichlet function has no limit?

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• 1. Besides Dirichlet functions, which functions are Riemann non integrable and Lebesgue integrable,
• 2. 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
• 3. 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)
• 4. 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
• 5. Why can't Dirichlet higher numbers be expressed as limit functions of continuous functions
• 6. 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?
• 7. Is Dirichlet function continuous almost everywhere on R? I know it's discontinuous everywhere. Is it continuous almost everywhere in real change?
• 8. 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
• 9. 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.
• 10. Write a number whose product with 2 √ 3 is a rational number
• 11. What are the common forms of periodic functions Excuse me?
• 12. 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!
• 13. What is the fundamental period of a function
• 14. 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
• 15. How to find the minimum value and the minimum positive period of a function
• 16. How to find the minimum positive period in positive metaphysical function?
• 17. 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
• 18. Two questions about the proof of periodic function 1. It is known that f (x) is an odd function, and the image of F (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function 2. Let f (x) be an even function defined on R whose image is symmetric with respect to the line x = 1. For any x 1, x 2 belonging to [0,0.5], f (x 1 + x 2) = f (x 1) * f (x 2) is proved to be a periodic function
• 19. How to prove the periodic problem of two functions? If f (x) is an odd function and the equation f (a + x) = f (A-X) holds for all x ∈ R, it is proved that the period of F (x) is 4a If f (x) is symmetric with respect to (a, Y0) and x = B, it is proved that the period of F (x) is 4 (B-A)
• 20. Can we give the operational properties of function period in detail For example, f (x) = f (x + T) or other In addition, is f (x-a) = f (x + a) a periodic operation?