Continuity of the derivative function.

Question

Suppose we have a function f(x) that is differentiable for all values of x. Is it necessary for the derivative function to be continuous for all x ?.

No, it is not. It is however necessary that the derivative satisfy the "intermediate value property": if f'(a)< u< f'(b) then there exist x, between a and b, such such that f'(x)= u. All continuous functions have that property but not all functions, having that property, are continuousl — user247327, Oct 23 '20 at 13:51
Can you give me an example of a function that has a derivative everywhere but the derivative function has a discontinuity ? — Ahmad Eldesokey, Oct 23 '20 at 13:57

score 2 · Answer 1 · answered Oct 23 '20 at 13:57

2

Let $$g(x) = \begin{cases} x^2 \sin(1/x) \text{ if } x \neq 0 \\ 0 \text{ if } x = 0.\end{cases}$$

(1) Show this function is continuous.

(2) Show it's differentiable everywhere

(3) Show the derivative is not continuous at $x = 0$.

answered Oct 23 '20 at 13:57

User203940

2,473

If you get stuck, look here: https://math.stackexchange.com/questions/292275/discontinuous-derivative – User203940 Oct 23 '20 at 13:58

score 1 · Answer 2 · edited Oct 23 '20 at 14:41

No, it is not. For example, let $f(x)= x^2\sin(1/x)$ for $x \neq 0$, $f(0)= 0$. $f$ is differentiable for all $x$. Its derivative is $f'(x)= 2x \sin(1/x)- \cos(1/x)$ if $x \neq 0$, $f'(0)= 0$ which is not continuous at $x= 0$.

It is however necessary that the derivative satisfy the "intermediate value property": if $f'(a)< u< f'(b)$ then there exists $x$, between $a$ and $b$, such such that $f'(x)= u$. All continuous functions have that property but not all functions, having that property, are continuous.

Continuity of the derivative function.

2 Answers2