Applications of Rolle's Theorem and function analysis

Question

I was playing around with the Rolle's theorem when I ended up fixing a tricky constraint for a function. Suppose, $f$ is a differentiable function in $[0,1]\to\mathbb R$ such that $f(0)=f(1)$. Now if I set a constraint like $f(x)\neq f(x+1/3)$ for all $x\in(0,2/3)$, then can I find such a function? Since $f(0)=f(1)$, there had to be a point where $f'(x)=0$ in $(0,1)$. Intuitively, I feel that there must be some $x\in(0,2/3)$ such that $f(x)=f(x+1/3)$, as it's many-one. But can someone help me prove this using Rolle's or Lagrange's?