Say a function f is differentiable. Prove whether its derivative is continuous over its domain or not.
On an intuitive level, I believe it should be always continuous over the domain (open intervals) and am thinking of a rigorous proof.
Asked
Active
Viewed 50 times
0
-
The important thing to know is this: Even when $f'$ is not continuous, $f'$ always has the intermediate value property. – Ted Shifrin Dec 22 '19 at 20:12
1 Answers
0
Let $f(x)=x^{2}\sin(1/x^{2})$, $x\ne 0$, $f(0)=0$, then $f'(0)=0$ and $f'(x)=2x\sin(1/x^{2})-(2/x)\cos(1/x^{2})$ for $x\ne 0$, not only that $f'$ is discontinuous, $f'$ is also unbounded.
user284331
- 56,315