Is there no limit if there are discontinuities?

The first kind of discontinuity has a limit

The limit of a function doesn't exist somewhere. Is it the discontinuity of a function somewhere,

Somewhere is the inequality of left and right limits, and beyond the domain of definition

Does the derivative of a function have the first kind of discontinuity?

There is no discontinuity of the first kind for the derivative function in its domain of definition, that is to say, the derivative function is defined at its discontinuity point (that is, the original function has derivative at this point), so this point can not be the discontinuity of the first kind for the reason that, if the derivative function is defined at this point (the original function can be derivative at this point), then it can not be the discontinuity of the first kind, The left and right derivatives of the original function at this point exist, but they are not equal. Then the left and right derivatives of the original function at this point are respectively equal to the left and feasible limits of the derivative function at this point. However, because these two limits are not equal, the left and right derivatives of the original function at this point are not equal, which is in contradiction with the definition of the derivative function at this point (the original function has derivatives at this point), Therefore, if the derivative function has left and right limits and is not equal at this point, then the derivative function is not defined at this point (the original function is not differentiable at this point, because the left and right derivatives are not equal). If the derivative function is required to have a definition at this point (the original function is differentiable at this point), then the two upper limits of the derivative function at this point are either equal, or at least one of them does not exist

[FAQ] Why "there is no discontinuity of the first kind in derivative function"
As the title of this problem has not been to understand

Let f (x) be differentiable at (a, b) and x0 belong to the discontinuous point of (a, b) f '(x). By way of counter proof, if the right limit of the first kind of discontinuous point F' (x) at x0 is a +, the left limit is A-and the right derivative of F (x) at x0 is a +, the left derivative is A-and because the derivative of F (x) at x0 exists, the right derivative of F (x) is a +, So the left derivative is equal to the right derivative is equal to f '(x0), the limit of F' (x) at x0 is equal to f '(x0), the continuity of F' (x0) at x0 is inconsistent with the known, so there is no first kind of discontinuity PS: f '(x) refers to the derivative of F (x), I'm afraid someone can't see it clearly

Let f (x) = (x ^ 2-1) / [| x | (x-1)], then the first kind of discontinuity is 0. Why

The answer is wrong, because this function is an elementary function, except 0 and 1, the function is continuous, the expression is (x + 1) / | x |, the limit at point x = 1 is 2, and the limit at point 0 is infinite, so x = 1 is the first kind of removable discontinuity of this function, and x = 0 is the second kind of infinite discontinuity of the function. If the answer is as you said, you must be wrong about the function expression

Don't use a calculator
Without a calculator, which is larger than √ 7-1 or 1.5?

1.5
=2.5-1
=√(2.5*2.5)-1
=√6.25-1

How to calculate the formula of covariance?
CoV (x, y) = e {[x-e (x)] [y-e (y)]}, where are e, e (x) and E (y)?

There is a Xi in the mean value of X, so the two are related
In order to make the two independent, we separate the components of Xi from the mean value of X, that is, j is not equal to I in the sum, and Xi of N in the sum formula is mentioned separately
Now we can expand the variance and do not need to find the covariance difference

How to calculate the covariance in personal finance?
For example: probability of economic situation a rate of return B rate of return
Depression 20% - 0.150.05
Recession 5% 0.20 0.10
Normal 40% 0.30 - 0.05
Prosperity 35% 0.450.15 seek the covariance of a B return

Expected rate of return of a
0.15*20%+0.20*5%+0.30*40%+0.45*35%=31.25%
B's expected rate of return
0.05*20%+0.10*5%+0.05*40%+0.15*35%=8.75%
covariance
20%*（0.15-31.25%）（0.05-8.75%）+5%*（0.20-31.25%）(0.10-8.75%)+40%(0.30-31.25%)(0.05-8.75%)+35%*(0.45-31.25%)(0.15-8.75)=
It's not easy. There are so many numbers. I strongly demand extra points

Covariance calculation formula derivation process. Thank you

Covariance: covariance represents the overall error of two variables. If the change trend of two variables is consistent, that is, if one of them is greater than its own expected value, and the other is greater than its own expected value, then the covariance between two variables is positive. If the change trend of two variables is opposite, that is, one of them is greater than its own expected value, If the other is less than the expected value, the covariance between the two variables is negative
Covariance formula: X and y are two random variables; cov (x, y) = E [(x-e (x)) (y-e (y))]

Calculation method of covariance
x = 3 2 4 5 6
y = 9 7 12 15 17
How to put in the formula

First, the expected values of X and y are calculated
ux=（3+2+4+5+6）/5=4
uy=（9+7+12+15+17）/5=12
Take Xi (3, 2, 4, 5, 6), Yi (9, 7, 12, 15, 17) and the expectation calculated above into the formula by using the formula you gave,
Cov（X,Y）=26/5.

Is there no limit if there are discontinuities?