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I have been thinking about this problem but I am not able to find a solution.

Ler h be a continuous function in the range [0, 2] and such that h (0) = h (2). Prove that there are at least two points a, b ∈ [0, 2] whose distance is equal to 1 and for which h (a) = h (b) is verified.

Best regardes.

Davide Severino
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  • See https://math.stackexchange.com/questions/3366262/let-f0-n-to-bbb-r-be-continuous-with-f0-fn-then-there-are-n-pairs for a more general question – Maximilian Janisch Mar 23 '21 at 21:47

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Hint: Consider $g\colon[0,1]\to\Bbb R$ given by $g(x)=h(x)-h(x+1)$.

Hagen von Eitzen
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  • what I’m not achieving is how I can make inferences for a and b because I have to prove that h (a) = h (b). The function g is true for h(0) and h(2) but i don’t see how can o use to prove for a and b. Can you help me? – Davide Severino Dec 12 '20 at 15:35