An example of a non-constant function with the following properties
Let $f: \mathbb{R} \to \mathbb{R}$ be a non-constant function with the following property: Given $\epsilon > 0$, there exists a positive real number $r > 0$ and a set $D \subset \mathbb{R}$ such that
- Any interval $I$ with length $r >0$, is such that $D \cap I \ne \emptyset$
- $|f(x+t) - f(x)| < \epsilon$ for any $t \in D$ and $x\in \mathbb{R}$
I'm having a hard time picturing a function that is non-constant that satisfies both properties. If we take $D = \mathbb{Q}$ then the first property is easily satisfied, but I can't figure out what kind of function satisfies the second property. Can someone give me an example?
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$\begingroup$Let $f(x)=\sin x$ and (no matter what $\epsilon$ is), let $r=2\pi$, $D=2\pi\Bbb Z$.
$\endgroup$ 6 $\begingroup$Consider the equivalence relation $\sim$ on $\Bbb{R}$ defined by $x \sim y$ iff $x - y \in \Bbb{Q}$ and let $P$ be the set of equivalence classes of $\sim$. Choose any non-constant function $g :P \to \Bbb{R}$ and define $f : \Bbb{R} \to \Bbb{R}$, by $f(x) = g([x])$ where $[x]$ denotes the $\sim$-equivalence class of $x$. Then with $D = \Bbb{Q}$ any interval $I$ of positive length meets $D$ and $|f(x + t) - f(x)| = 0$ for any $t \in D$.
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