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 $f(x+\theta)-f(x)\ne0$

Since I cannot close as dupe because of the bounty.

Here is prior art: Universal Chord Theorem

Tags:

Analysis

Calculus

Functions

Real Analysis

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