TITLE: For $0\lt\theta\lt1$, $\frac 1\theta\notin\mathbb Z$, there exists $f\in C[0, 1]$ such that $f(0)=f(1)$ and $f(x+\theta)-f(x)\ne0$
QUESTION [5 upvotes]: Prove that for each $0\lt\theta\lt1, \dfrac{1}{\theta} $ isn't an integer, there exists $f \in C[0, 1]$ such that $f(0)=f(1)$, and $ \forall x\in[0,1-\theta]  , f(x+\theta)-f(x)\ne0 $
(If $\theta\gt\frac12, $ it is obvious that such an $f$ exists.)

REPLY [2 votes]: Since I cannot close as dupe because of the bounty.
Here is prior art: Universal Chord Theorem