Unit fraction chord lengths

Another interesting property of unit fractions (numbers of the form $\dfrac{1}{n}$ for positive integer $n$).

Part A)

Suppose $f:[0,1] \to \mathbb{R}$ is any continuous function such that $f(0) = f(1)$.

Show that for every natural number $n \gt 0$, there is some $c \in [0,1]$ ($c$ dependent on $n$ and $f$) such that $f(c) = f\left(c + \dfrac{1}{n}\right)$

Basically the graph of $f$ has a chord of length $\dfrac{1}{n}$.

Part B)

Suppose $r \in (0,1)$ is a real number such that $r$ is not a unit fraction.

Show that there is some continuous function $f_r$ (dependent on $r$) such that

$f_r:[0,1] \to \mathbb{R}$, $f_r(0) = f_r(1)$ and for any $c \in [0,1-r]$, $f_r(c) \ne f_r(c+r)$.

i.e the graph of $f_r$ has no chord of length $r$.

[Solution]