Simple dirichlet function differentiability

In summary, in order to show that f(x) is not differentiable at 0, we can use the definition of differentiability and the fact that f(x) is defined differently for rational and irrational values. By taking the limit of the difference quotient, we can see that the limit is undefined as h approaches 0, proving that f(x) is not differentiable at 0. On the other hand, for g(x), we can use the same approach to show that it is differentiable at 0, as the limit of the difference quotient is equal to 0.
  • #1
Nanatsu
3
0

Homework Statement


f(x) = {x, x rational, 0, x irrational

g(x) = {x^2, x rational, 0, x irrational

Show that f(x) is not differentiable at 0.
Show that g(x) is differentiable at 0


Homework Equations


f'(x) = lim(h->0) f(x+h) - f(x)/h I suppose


The Attempt at a Solution


Just wondering if I'm thinking right. For f(x) the difference quotient becomes h/0 as h-> 0 and with g(x) it becomes lim(h->0) h^2/h = lim(h->0) h = 0?
 
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  • #2
"
Just wondering if I'm thinking right. For f(x) the difference quotient becomes h/0 as h-> 0

I suppose you mean h/h?

and with g(x) it becomes lim(h->0) h2/h = lim(h->0) h = 0?"

That's the general idea, but you have to consider the fact that h may be rational or irrational in a careful writeup.
 

FAQ: Simple dirichlet function differentiability

1. What is the simple dirichlet function?

The simple dirichlet function is a mathematical function that is defined as follows: $$f(x) = \begin{cases} 1 & \text{if } x \text{ is irrational} \\ 0 & \text{if } x \text{ is rational} \end{cases} $$This means that the function outputs a value of 1 if the input is an irrational number, and a value of 0 if the input is a rational number.

2. What is the purpose of the simple dirichlet function?

The simple dirichlet function is often used in mathematics as a simple example of a function that is discontinuous everywhere. It has no defined derivative at any point, making it a useful tool for studying differentiability and continuity.

3. How is the simple dirichlet function differentiable?

The simple dirichlet function is not differentiable at any point, as it is discontinuous everywhere. This means that it does not have a defined derivative at any point, and therefore is not considered differentiable.

4. Are there any real-world applications of the simple dirichlet function?

The simple dirichlet function does not have many direct real-world applications, as it is a purely mathematical concept. However, it is used as a tool in studying differentiability and continuity, which have many real-world applications in fields such as physics, engineering, and economics.

5. How is the simple dirichlet function related to the dirichlet function?

The simple dirichlet function is a simplified version of the dirichlet function, which is defined as follows: $$f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases} $$Both functions have similar properties, such as being discontinuous everywhere and having no defined derivative at any point. However, the dirichlet function is slightly more complex and is used in various fields of mathematics, such as number theory and analysis.

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