Differentiable continuous function whose derivative is not continuous

Question

Is there a function which is continuous and differentiable, but is not smooth function?

By smooth I mean having continuous derivative. For example, the derivative of $f(x)=x|x|/2$ is $f'(x)=|x|$ which is continuous. So I consider this function smooth.

To have continuous derivatives. The derivative of x|x|/2 is |x| which is continuous. So I think it's smooth — user96634, Jan 14 '15 at 19:21
To follow-up on Git Gud's comment, there are many different uses of "smooth". For instance, one of the theorems in my Ph.D. thesis involves smooth functions (more precisely, functions that are non-smooth at each point), but I bet the notion is different than what you're asking about. — Dave L. Renfro, Jan 14 '15 at 19:21
@DaveL.Renfro I didn't know about that, I thought smooth could only mean infinitely differentiable. — Git Gud, Jan 14 '15 at 19:25
@user96634 In view of Dave's comment above, please add the definition of smooth to the question. — Git Gud, Jan 14 '15 at 19:25
Also of http://math.stackexchange.com/questions/292275/discontinuous-derivative — , Jan 14 '15 at 19:29
@Git Gud: Off-hand, I can think of three different notions I've seen "smooth" refer to, and if I were sufficiently inclined to (which I'm not), I suspect I could find one or two more if I dug around for a couple of days. The three notions I can think of are continuously differentiable, infinitely differentiable, and what I call (to avoid ambiguity) "Zygmund smooth", the last of which can be found here among many other places. — Dave L. Renfro, Jan 14 '15 at 19:34

score 7 · Accepted Answer · answered Jan 14 '15 at 19:36

7

One standard example is $f(0) = 0$, $f(x) = x^2*\sin(1/x)$. Then $f'(0) = 0$, but $f'$ is not continuous at $0$.

answered Jan 14 '15 at 19:36

Tomas Persson

507

Differentiable continuous function whose derivative is not continuous

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