Lipschitz Q: Show w/ Example & Derivative

[JasMath33]

Homework Statement

Homework Equations

The Attempt at a Solution

I know I will just have to show this by one example. I thought about using f(x) = x² but I am not sure if this satisfies the last part dealing with the absolute value of the derivative. It is just the last part on which I am stuck.

 

[Ray Vickson]

Homework Statement

View attachment 102885

Homework Equations

The Attempt at a Solution

I know I will just have to show this by one example. I thought about using f(x) = x² but I am not sure if this satisfies the last part dealing with the absolute value of the derivative. It is just the last part on which I am stuck.

Just showing it for one example will not satisfy the requirements of the question.

 

[JasMath33]

Just showing it for one example will not satisfy the requirements of the question.

But I don't need to prove it for all functions. Why would not showing one work?

 

[Ray Vickson]

But I don't need to prove it for all functions. Why would not showing one work?

You have mis-read the question. It hypothesized some properties of an uspecfied, general function $f(x)$ and asked you to prove something else about that function. Showing it for just one function alone won't work; how do you know it would be true for some other function that you did not use?

Of course, showing it for one function privately (for your own background use only) may help you to develop the needed intuition about the problem, thus allowing you to extend the ideas to the general case, but doing the general case is absolutely required.