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It is easy to prove that if $f(x)$ is periodic with period $T$,then $f'(x+T)=f'(x)$. However I don't know how to prove $T$ is the shortest period for $f'(x)$.

Can anyone help me? Or if it is not true, can anyone give me a counterexample?

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maybe this helps – dato datuashvili Aug 7 '12 at 8:24
up vote 1 down vote accepted

It seems that the hint I have given before was too obscure. So I now give a full solution.

By assumption $f'$ is the derivative of a $T$-periodic function $f$, where $T>0$ is a primitive period of $f$. Therefore $f'$ has period $T$ as well.

If $f'$ were a constant then this constant would be $0$ (or $f$ would not be periodic). As a consequence $f$ would be constant and would not have a primitive period.

Therefore $f'$ has a primitive period $p>0$, which then necessarily is of the form $p={T\over n}$ for some $n\geq1$. Put $\int_0^p f'(t)\ dt=:c$. Then $$\int_x^{x+p}f'(t)\ dt=c$$ for any $x$; furthermore one has $$0=f(T)-f(0)=\int_0^T f'(t)\ dt=n\int_0^p f'(t)\ dt=n\, c\ .$$ Therefore $c=0$, and this in turn implies that $$f(x+p)-f(x)=\int_x^{x+p} f'(t)\ dt=0$$ for all $x$; whence $f$ has period $p$. This implies $p= m T$ for some $m\geq1$, so that we in fact have $p=T$.

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If $f^{'}(x)=1$? – 89085731 Aug 7 '12 at 8:39
then $f(x)=x+c$ ; it would have period $p$ as well if $c=0$ – dato datuashvili Aug 7 '12 at 8:47
@dato what is the period of x – 89085731 Aug 7 '12 at 8:50
sorry when $x+c=1$ – dato datuashvili Aug 7 '12 at 8:58

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