Proving continuity of binary function by differentiability How to prove Just one point is enough

It's very easy for you to deform the limit formula. What we can say is that (I use DX, Dy to express the increment of X, y), there are numbers a and B such that LIM (DX, Dy tends to zero) [f (x0 + DX, Y0 + dy) - F (x0, Y0) - ADX - bdy] / sqrt [(DX) ^ 2 + (Dy) ^ 2] = 0, because f (x0 +...)

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1. The problem of continuity and differentiability of binary functions 1. F (x, y) - f (0,0) + 2x-y = O (ρ), (when (x, y) → (0,0)), we can get that f (x, y) is differentiable at point (0,0). Why? How? 2. LIM (x, y) → (0,0) (f (x, y) - f (0,0) + 2x-y) = 0 can get f (x, y) continuous at point (0,0). Why? How?
2. Give m exercise books to n students, if each student has 3 exercise books, then the remaining 80; if each student has 5 exercise books, then the last student has less than 5 exercise books, and the value of n is --- 41 or 42 The result is n > 40, why 41 or 42 The formula is as follows: from the meaning of the title, 3n+80=m m-5（n-1）
3. The unit price of apple per kilogram is 2.5 yuan. Please write down the analytic formula and domain of the function between the total price of apple and the weight x kilogram Just taught me how to learn function. Thank you
4. Xiao Gang bought an exercise book with 3 yuan. Given that each book costs 0.25 yuan, he wrote out the functional relationship between the number of books X and the remaining money y, and made the function image
5. The unit price of exercise books is 0.6 yuan. The relationship between the quantity x (yuan) of exercise books and the total price y (yuan) (about function)
6. Let f (x) have a third derivative. When x tends to x0, f (x) is the second order infinitesimal of x-x0. What are the characteristics of Taylor expansion of F (x) at x0? In addition, find out what Lim f (x) / (x-x0) ^ 2 is equal to when x tends to x0
7. Why is the limit of infinitesimal not negative and 0?
8. Why is it that when n tends to infinity, the n-th sine divided by n-th one equals 1? Is the infinitesimal divided by the infinitesimal equal to one?
9. How to find LIM (x tends to 0 +) LNX / 2x?
10. Finding limit limx-0 sin3x / sin5x
11. When the value of independent variable x is, the value of function y = 3x-15 is less than 0? When the value of independent variable x is, the value of function y = 2X-4 is greater than 0=
12. Given the function y = f (2x-1), ask whether the independent variable is x or 2x-1? Given the domain of y = f (2x-1), how to find the domain of F (x)? Given the domain of y = f (x), how to find the domain of F (2x-1)?
13. When the value of independent variable x satisfies what conditions, the value of function y = 3 / 2x + 6 satisfies the following conditions. Y = 0 y < 0 Y > 0 y < 2 When the value of independent variable x satisfies what conditions, the value of function y = 3 / 2x + 6 satisfies the following conditions y=0 y＜0 y＞0 y＜2
14. How to make multiple judgments for a function formula in electronic form Fill in the corresponding number in A1 and B1 to get C1, and the number range of C1 should conform to a value shown in D1. (the formula of C1 has been filled in) ten
15. How can you think of using the definition of infinity and infinitesimal when you see the question I asked
16. Is limit equal to 0 equal to infinitesimal
17. Infinitesimal in limit problems of higher numbers In the limit of higher numbers, the definition of infinitesimal is f (x) when x approaches to x0 or when the limit is zero, then f (x) is called the infinitesimal of X in this process. However, in the related proof later, it seems that the definition of infinitesimal when x approaches to x0 appears again. By definition, isn't infinitesimal only a function? Why can it be replaced by a specific symbol (constant?)? Thank you~
18. What is the product of infinitesimal and infinitesimal? Such as the title
19. Given the function f (x) = − 14x4 + 23x3 + AX2 − 2x − 2, it decreases monotonically in the interval [- 1,1] and increases monotonically in the interval [1,2]. Find the value of real number a
20. Given the function f (x) = x ^ 3-x decreasing on (0, a) and increasing on [a, positive infinity), try to find the value of A