True or False :
The derivative of a periodic functions is always periodic.
I thought it to be true , as everything about a periodic function repeats itself at regular intervals, and so should it's derivative . But , to my surprise it is given false , which suggests that it might be true most of the time but not always , I have given all my thoughts to finding a counter example but I just can't seem to find even one counter example .
One possibility was $\{x\}$ , which is not differentiable at every integer , and I am confused about whether I should call it periodic or not , because it's graph will be a straight line with holes at every integer , so in a sense it is periodically not defined , just like $\tan x$ wich is not defined at every odd multiple of $\pi\over 2$ but still it is said to be periodic .
Could someone please help me find a counter example and clarify about periodicity of derivative of $\{x\}$.
Thanks !
$\{x\}$ is fractional part of x .
Best Answer
If $f$ is periodic with period $T$, then $f'$ is also periodic with period $T$, because, if $f$ is differentiable at $x$,\begin{align}f'(x+T)&=\lim_{h\to0}\frac{f(x+T+h)-f(x+T)}h\\&=\lim_{h\to0}\frac{f(x+h)-f(x)}h\\&=f'(x).\end{align}And $\{x\}'$ is periodic with period $1$ (although its domain is not $\Bbb R$).