here is a question from calculus exam that I have been struggling with:
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that the set: $$A = \{ T > 0 : f(x + T) = f(x) \text{ for all } x \in \mathbb{R} \}$$
is nonempty (i.e.,
f is periodic with some period).
prove that $f$ is constant if and only if $\inf A = 0$
So to prove that if $f$ is constant then $\inf A = 0$ is easy, I am struggling with the second direction.
Here is my intuition for the second direction:
We will show that if $f$ isn't constant then $\inf A$ is not $0$.
We should show that if $f$ is not constant then A has minimum, and because for every $T\in A. T>0$ Then $\inf A \not= 0$.
We know that $A$ is nonempty group so there is a $T\in A$ such as $f(x+T)=f(x)$
from here I don't have any clever thing to say.
