Is there exists a continuous function whose derivate function is NOT continuous? To be more specific, $f$ is continuous while $f'$ isn't. I'm just looking for a drawable example for a project. Thanks in advance.
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How about the absolute value function? $$ f(x) = |x| $$ – Matti P. Feb 10 '20 at 12:41
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continuous, not continues. – Feb 10 '20 at 12:41
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@ Matti: the absolut value function is not differentiable at $0$. – Fred Feb 10 '20 at 12:43
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@MattiP.: undefined at $x=0$. – Feb 10 '20 at 12:43
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Example: $f(x)=x^2 \sin(1/x)$ for $x \in (0,1]$ and $f(0)=0.$
$f$ is continuous on $[0,1]$,
$f$ is differentiable on $[0,1]$,
$f'$ is not continuous at $0$.
Fred
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thank you so much, it seems understandable. Sorry but can you think of any other example? And basically what gives the function that adjective? – Boshra Alef Feb 10 '20 at 12:55