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I saw a question and there was a comment who said there is not such function, but I was not able to se how, any hint?

Find map $f : \mathbb{R}\to {\mathbb{R}}$ such that inverse image $f^{-1}(${$x$}$)$ has two elements for any $x\in \mathbb{R}$.

Sumanta
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George
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    Check this: https://math.stackexchange.com/q/677085/42969 or this: https://math.stackexchange.com/q/1673609/42969 – Martin R Nov 26 '21 at 21:20
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    Alternatively you can ask "Does there exist a continuous function such that any horizontal line $y=c$ for some constant $c$ intersects exactly two points? Then ask what happens when monoticity changes in the function. What does that imply when the function is continuous? – CyclotomicField Nov 26 '21 at 21:23
  • Why is this tagged "algebraic-topology"? – John Palmieri Nov 27 '21 at 01:11

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