How to prove that Riemann function is not differentiable everywhere

How to prove that Riemann function is not differentiable everywhere

Just by definition
The derivatives at rational and irrational points are discussed respectively

A differentiable function must be continuous. A discontinuous function must not be differentiable Which number does the function here refer to? How do you know which functions are between 1 and N? For example..

In mathematics, a function is a relationship that makes each element in one set correspond to a unique element in another (possibly the same) set. A function does not refer to a specific number
For example:
Sine function: y = SiNx
Cosine function: y = cosx
Where x is the independent variable and Y is the dependent variable
If you draw a graph, the above two function lines are not disconnected, smooth, and have no edges and corners. This is what differentiable functions look like. Although the lines of continuous but non differentiable functions are connected from beginning to end, they are not smooth, and have edges and corners. You can touch them with your hand

It is proved that when the function y = f (x) is differentiable at point X., then f (x) must be differentiable at point X

Let me help you
If the function f (x) is differentiable at x0
Then, defined by differentiability, for the function, the variable △ y,
With △ y = a △ x + O (△ x)
Where a is independent of △ x, and O (△ x) is the higher-order infinitesimal of △ X
Divide △ x on both sides, and then take the limit at the same time
There is Lim △ Y / △ x = Lima △ X / △ x + limo (△ x) / △ X
=A+0=A
So the limit exists. (LIM △ Y / △ x exists, which is the derivative definition)
So it's differentiable by x0
Note: △ x is the independent variable divided by x0, and △ X - > 0

How to prove that the function y = |x| is continuous and non differentiable at x = 0 Why is "function at x = 0, left limit = 0, right limit = 0, both = f (0), so; continuous"? Also, what is the definition of the limit of a function and why I don't see it in my textbook. For example, how to find the limit of this question? The left and right limits are equal and equal to the function value there, so they are continuous. Why?

The necessary and sufficient condition for a function to be continuous is that the left and right limits exist and are equal to its function value y = |x|. When x > 0, y = x, X tends to 0 +, y equals 0, y '= 1. When x < 0, y = - x, x tends to 0 -, y equals 0, y' = - 1. Because x = 0, y = 0, it is continuous, but the left and right derivatives are different. Therefore, the definition of the limit of a non differentiable function is when the independent variable

It is proved that if the function y = f (x) is continuous in a, and f (a) ≠ 0, and the function [f (x)] ^ 2 is differentiable in a, then the function f (x) is also differentiable in a

The function [f (x)] ^ 2 is known to be differentiable at x = a, that is, the limit
　　　　lim(x→a)[f ² (x)-f ² (a)]/(x-a) = A
Exists, and f (x) is continuous at x = a, and f (a) ≠ 0, so
　　　　lim(x→a)f(x) = f(a),
therefore
　　　　lim(x→a)[f(x)-f(a)]/(x-a)
= lim(x→a){[f ² (x)-f ² (a)]/(x-a)}*{1/[f(x)+f(a)]}
= A*[1/2f(a)]
= A/2f(a),
By definition, f '(a) exists, and
f'(a) = C/2f(a).

Prove that the function y = {SiN x, x < 0; x. X ≥ 0. Differentiable at x = 0

Slope of y = x = 1 (at x = 0)
Y = slope of SiNx = 1, so the left and right tangents are the same