There exists at least one $x \in [0, 1 - \frac{1}{n}]$ such that $f(x) = f(x + \frac{1}{n})$.

There exists at least one $x \in [0, 1]$ such that $f(x) = 0$.

The function $f$ must be constant on $[0, 1]$.

The derivative $f'(x)$ must be zero for some $x \in (0, 1)$.