If $f(x)$ is decreasing, differentiable, and positive for $x\ge1$, must $f'(x)$ be continuous for $x\ge1$?

Asked Mar 26 '16 at 16:07

Active Mar 26 '16 at 16:14

Viewed 66 times

If $f(x)$ is decreasing, differentiable, and positive for $x\ge1$, must $f'(x)$ be continuous for $x\ge1$?

In general, what conditions on a real-valued differentiable function $f$ guarantee that its derivative is continuous?

edited Mar 26 '16 at 16:14

egreg

asked Mar 26 '16 at 16:07

clburchard

3

Let $f(x)=4x+x^2\sin(1/x)$, $x\ne0$; $f(0)=0$. $f$ is differentiable and monotone on $(-1,1)$, but $f'$ is not continuous at $x=0$. Appropriate translations/reflections of $f$ should give a counterexample. – David Mitra Mar 26 '16 at 16:25
For your second question, you might find this of interest. – David Mitra Mar 26 '16 at 16:38
@DavidMitra Wow - just that one counterexample is food for thought. My obviously suspect intuition wouldn't have had me believing that that derivative could be discontinuous on the interval. Thanks you. – clburchard Mar 26 '16 at 20:52

0 Answers0