Periodic Function with no Minimum is Constant

Question

Let $f: \mathbb R \rightarrow \mathbb R$ be a periodic function, which means that there is some positive $p$ such that $f(x)=f(x+p)$ for all $x$. Is it the case that, if there is no minimum such $p$, then the $f$ must be a constant function?

first of all if there is no minimum of $p$,then this function is infinity periodic or aperiodic is not it? — dato datuashvili, Jan 25 '14 at 07:08

score 5 · Accepted Answer · answered Jan 25 '14 at 07:09

5

No: Let $f$ be the function which is $1$ on all the rationals, and $0$ on all the irrationals. Then every positive rational number is a period of $f$.

answered Jan 25 '14 at 07:09

what if we require f to be cont. ? – voldemort Jan 25 '14 at 07:44
1

@voldemort: Let $f(0)=c$. For any $a$, there are $x$ arbitrarily close to $a$ such that $f(x)=c$. But if $f$ is continuous at $a$, we have $f(a)=\lim_{x\to a}f(x)=c$. – André Nicolas Jan 25 '14 at 08:07

score 1 · Answer 2 · answered Jan 25 '14 at 07:09

1

I found a kind of explanation on Math-World,"The constant function f(x)=0 is periodic with any period R for all nonzero real numbers R, so there is no concept analogous to the least period for constant functions."

answered Jan 25 '14 at 07:09

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Periodic Function with no Minimum is Constant

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