Continuous Function and use of IVP

Question

Let $\;f:[0,1]\to[0,1]\;$ be continuous function such that $f(0)=f(1)$ and let $A =\{t,s ∈ [0,1]\times[0,1]\; |\; t \neq s ,\;f(t) =f(s) \}$ then prove that number of elements in A is infinite. I feel I have to prove it a constant function using intermediate value property but not getting articulate way of writing the proof .

Answer 1

Hint : The argument goes as follows.

Let $$ M=sup_{t\in [0,1]}f(t)\ ;\ m=inf_{t\in [0,1]}f(t) $$ then $m\leq f(0)\leq M$ and discuss the cases

1. $m=f(0)=f(1)=M$
2. $m<f(0)=f(1)$
3. $f(0)=f(1)<M$

Further : In case (1), $A=([0,1]\times [0,1])\setminus \Delta$ ($\Delta$ is the diagonal i.e. $\Delta=\{(x,x)\}_{x\in [0,1]}$) so it is infinite. In case (2), note $x_m$ a point such that $f(x_m)=m$, now, by IVP for each $m<c<f(0)$, the equation $f(x)=c$ has at least one solution in $]0,x_m[$ (say $a_c$) and one in $]x_m,1[$ (say $b_c$). One has $\{(a_c,b_c)\}_{m<c<f(0)}\subset A$ which is, therefore, infinite. Try to write case (3).

Hope it helps. Do not hesitate.

PS : I do not agree with the downvote(s) as this question, even trivial to visualize, needs some care to write down.