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Does there exist a continuous function f : R → R that takes every real value exactly twice?

What is the meaning of taking value exactly twice?

Himanshu
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    It means $f(x)=k$ has two solutions for every $k$. – DHMO Apr 14 '17 at 07:26
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    It means that $f(x)=k$ has exactly two solutions for every $k$, to be precise. The fact that it must be exactly two, is essential to this question. – wythagoras Apr 14 '17 at 07:27
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    It is quite intuitive that there cannot exist such functions, as, intuitively speaking, they would have to go to $-\infty$ twice and $\infty$ twice, and continuous functions can only go to them once, or one of them twice. – DHMO Apr 14 '17 at 07:29
  • Apologies for hurried question reading. It means $f(x)=k$ has two solutions for every $k$ as @DHMOmentioned. – The Dead Legend Apr 14 '17 at 07:30

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