calculus - Find a Continuous Function

I am having a problem with this exercise. Could someone help?

Suppose $a \in (0,1)$ is a real number which is not of the form $\frac{1}{n}$ for any natural number n
n. Find a function f which is continuous on $[0, 1]$ and such that $f (0) = f (1)$ but which does not satisfy $f (x) = f (x + a)$ for any x with $x$, $x + a \in [0, 1]$.

I noticed that this condition is satisfied if and only if $f(x) \geq f(0)$

Thank you in advance

Answer

Look at $f(x) = \sin(2\pi x)$. For which values of $a$ can you find an $x \in [0,1]$ with $x+a \in [0,1]$ and $f(x) = f(x+a)$? In particular, if you additionally require $a > \frac{1}{2}$, can such an $a$ exist at all?

Once you've answered that you've solved your problem for some values of $a$. Which are those?

A general solution can be found in the answers to Universal Chord Theorem (Link found by the user who asked the question). To quote

$$f(x) = \sin^2\left(\frac{\pi x}{a}\right) - x \ \sin^2\left(\frac{\pi}{a}\right)$$

is a solution. This works because $f(x) = f(x+a)$ implies $a \sin^2\left(\frac{\pi}{a}\right) = 0$ and thus $a=\frac{1}{n}$ for some $n \in \mathbb{N}$. The answers to the linked questions also prove that $a \neq \frac{1}{n}$ for every $n \in \mathbb{N}$ is a necessary condition for a solution to exist.