Continuity of the derivative of a decreasing differentiable function

[Petek]

Homework Statement

To solve a problem in a book, I need to know whether or not the following is true:

Let f be a real-valued, decreasing differentiable function defined on the interval $[1, \infty)$ such that $\lim_{x \rightarrow \infty} f(x) = 0$. Then the derivative of f is continuous.

Homework Equations

N/A

The Attempt at a Solution

Tried working directly with the definitions (decreasing function, continuity, derivative), but got nowhere. Consulted various references (such as baby Rudin and Hobson's Theory of Functions of a Real Variable), with no luck. Spent time Googling combinations of the various terms, again with no results.

I suspect that the above statement is true. The canonical example of a differentiable function whose derivative is not continuous (see here) is not monotone in any interval containing the origin. However, I can't find either a proof or a counterexample.

 

[Dick]

Homework Statement

To solve a problem in a book, I need to know whether or not the following is true:

Let f be a real-valued, decreasing differentiable function defined on the interval $[1, \infty)$ such that $\lim_{x \rightarrow \infty} f(x) = 0$. Then the derivative of f is continuous.

Homework Equations

N/A

The Attempt at a Solution

Tried working directly with the definitions (decreasing function, continuity, derivative), but got nowhere. Consulted various references (such as baby Rudin and Hobson's Theory of Functions of a Real Variable), with no luck. Spent time Googling combinations of the various terms, again with no results.

I suspect that the above statement is true. The canonical example of a differentiable function whose derivative is not continuous (see here) is not monotone in any interval containing the origin. However, I can't find either a proof or a counterexample.

Your canonical example may not be monotone near the origin, but if you change a bit you can make one that is. Suppose you subtract say, 3x?

 

[Petek]

Thanks for the suggestion! By subtracting 3x from the function, we subtract 3 from the derivative. That makes the derivative negative and so the original function is monotone decreasing. I'll have to think about how to apply that to my original question.