# Monthly Archives: May 2017

## Counterexamples in differentiation

Exercise. Prove or disprove. If it is not true, find a counterexample.

1. and , then .
2. If is differentiable at , or are differentaible at .
3. If and are infinitely differentiable on and on , then or .
4. If is differentiable at and is differentiable at , then is differentiable at .
5. If is differentiable at , then is continuous at
6. Let . If for all , then is monotone increasing on .
7. If a function satisfies for all , then is constant on .
8. If is differentiable on and monotone increasing, then is monotone on .
9. If is bounded and is differentiable on and uniformly continuous on , then is bounded on .
10. If is differentiable on and , then .
11. If is differentiable on , then there exists such that

12. If and are continuous on and differentiable on and , then

13. Suppose that , and is bounded near . Then is local maximum.

Solution. (13)번만 제외하고 모두 거짓입니다. (13)만 제외하고 반례를 적어두겠습니다. 학생분들은 반례가 왜 명제에 어긋나는지 정확하게 기술해야 합니다.

(1) .

(2)

(3)

(4) , .

(5) if .

(6) ,

(7) . Define by if and if .

(8)

(9) , .

(10)

(11) Define by

(12) Define , .

(13) Assume is bounded on for some , say for all . For , by Taylor theorem, we have for some between and .

Note

Then

Choose so small that . This is possible since . So there is with such that for all .

Similarly, for the case , there is with such that for all . Choose . Then for , we have . This shows is a local maximum.