Let $I$ be an interval of $\mathbb{R}$ and $f$ a continuous function defined from I to $\mathbb{R}$ such that: $$(\forall x\in I)\quad (f(x))^{2}=1$$

Show that : $(\forall x\in I)\quad f(x)=1 \quad \mbox{or} \quad (\forall x\in I)\quad f(x)=-1$


closed as off-topic by Martin R, TheSimpliFire, Mostafa Ayaz, A. Goodier, Ove Ahlman Mar 1 '18 at 11:31

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  • $\begingroup$ Hint: Proof by contradiction and use Intermediate value theorem to find a root of $f$ in $I$, contradicting the condition $f^2(x) = 1$. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Feb 28 '18 at 12:43
  • $\begingroup$ So your $f^2(x)$ is the iterated application $f(f(x))$, not $f(x)$ squared? – In that case the statement would be wrong. $\endgroup$ – Martin R Feb 28 '18 at 12:44

I think that $f^2(x)$ means $(f(x))^2$ and not $f(f(x))$.

Suppose to the contrary, that there are $u,v \in I$ such that $f(u)=-1$ and $f(v)=1$. The intermediate value theorem gives now a point $w \in I$ with $f(w)=0$, hence

$$0 =f^2(w)=1,$$

which is absurd !


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