4
$\begingroup$

Can you give an example of a function $f(x)$ which is differentiable but not continuously differentiable and there exists a nbd $N$ around a point $c$ such that $f'(x) >(<) 0 \forall x \in N $ and $f'(x)$ is not continuous at the point $c$?

I am basically trying to search a function which is monotone in an open interval but it is not continuously differentiable at that interval.

Can anyone please help me to find that kind of a function?

$\endgroup$
  • 1
    $\begingroup$ Duplicate of 285944 $\endgroup$ – almagest Sep 25 '19 at 6:43
  • $\begingroup$ A function $g:\Bbb R\to \Bbb R$ can be increasing and differentiable and such that $\{f'(x): |x|<r\}$ is unbounded above for every $ r>0.$ Then $f(x)=x+g(x)$ is strictly increasing and differentiable and $f'(x)=1+g'(x)\ge 1$ for all $x,$ but $f'(x)$ is discontinuous at $x=0.$ $\endgroup$ – DanielWainfleet Sep 25 '19 at 11:08
3
$\begingroup$

$f(x) =10x + x^2 Cos(1/x) \forall x \neq 0$ , $0$ otherwise

.

take $N = (-0.5 , 0.5)$ and $f'(0) =10 , f'(x) > 0 \forall x \in N$ And $f'(x)$ is not continuous at $0$ .

| cite | improve this answer | |
$\endgroup$
3
$\begingroup$

Let $$g(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0.\end{cases}$$ Then $$g'(x) = \begin{cases} 2x \sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0.\end{cases},$$ hence $g'$ is discontinuous at $x = 0$. Note that, on the interval $(-1, 1)$, we have $$|g'(x)| \le 2|x| \cdot\left|\sin\left(\frac{1}{x}\right)\right| + \left|\cos\left(\frac{1}{x}\right)\right| \le 3.$$ Hence, if we let $$f(x) = 4x + g(x),$$ then $f'(x)$ is discontinuous at $x = 0$, and on the interval $(-1, 1)$, we have $$f'(x) = 4 + g'(x) \ge 4 - |g'(x)| \ge 4 - 3 > 0.$$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.