- #1
Petek
Gold Member
- 403
- 26
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 [itex] [1, \infty)[/itex] such that [itex]\lim_{x \rightarrow \infty} f(x) = 0[/itex]. 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.