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Question: Find all possible continuous functions $f:[0,1]\to \mathbb{R}$ such that $\forall x \in [0,1]$ we have $(f(x))^2=1$

My Work: I truly am not sure how to go about this. I don't quite understand what they mean by find all continuous functions. Any help on explaining what this question is asking would be great. Thanks

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    If $(f(x))^2=1$, then $f(x)$ has to be $\pm 1$ right? Then, ask yourself if it can be $1$ for some $x$'s and $-1$ for others while also being continuous. This should lead you to the answer.2012-11-12
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    Since $[0,1]$ is connected, if $f$ is continuous, then $f[0,1]$ must be connected too. The constraint implies $f(x) = \pm 1$ for all $x$, hence it must take exactly one of these values (otherwise the range in not connected).2012-11-12

4 Answers 4