# Continuity of the derivative of a decreasing differentiable function

1. Oct 24, 2012

### Petek

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

N/A

3. 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.

2. Oct 24, 2012

### Dick

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?

3. Oct 25, 2012

### 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.