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.

  • 3
    $\begingroup$ Feels like an Intermediate Value Theorem problem to me. $\endgroup$ – mweiss Aug 8 '16 at 4:56
  • $\begingroup$ Yes, IVT can be used, how we are going to relate c with c+1, $\endgroup$ – onurcanbektas Aug 8 '16 at 4:58
  • $\begingroup$ Formatting tips here. $\endgroup$ – Em. Aug 8 '16 at 8:57

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.


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

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


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