True or false: If $f(x)$ is continuous on $[0, 2]$ and $f(0)=f(2)$, then there exists a number $c\in [0, 1]$ such that $f(c) = f(c + 1)$.

Question

I was solving past exams of calculus course and I've encounter with a problem, which I couldn't figure out how to solve.The question is following;

Prove that whether the given statement is true or false:
Suppose that $f(x)$ is continuous on $[0, 2]$ and $f(0)=f(2)$. Then there exists a number $c\in [0, 1]$ such that $f(c) = f(c + 1)$.

By just reading the question, you want to say "use Mean Value Theorem or Rolle's theorem" but we don't know whether $f(x)$ is differentiable on the interval, at least it is now given, so I am totally stuck.I would appreciate any help.

Answer 1

Define a new function by $g(x)=f(x+1)-f(x)$. Notice that $$g(0) = f(1) - f(0)$$ $$g(1) = f(2) - f(1) = f(0) - f(1)$$ and therefore $g(0) = -g(1)$.

Now use the Intermediate Value Theorem.

Answer 2

Hint: Consider $\phi(x) = f(x+1)-f(x)$ on $[0,1]$.

Note that $\phi(1) = - \phi(0)$.