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This is the Intermediate Value Theorem as stated and proved in Munkres: Topology A First Course

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I understand everything except the claim that $(-\infty, r) \subset Y$ and $(r, +\infty) \subset Y$, I don't see why $-\infty, \infty \in Y$, since all we know about $Y$ is that it is an ordered set, we don't know anything about the structure or the elements of $Y$.

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    Where do you see that claim? – 2017-02-11
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    (replace $(-\infty, r)$ with $\{y\in Y: y< r\}$ and $(r,\infty)$ with $\{y\in Y:y>r\}$ if you have trouble with the notation) – 2017-02-11

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In an ordered space $(S,<)$, the order topology is generated by the subbase consisting of sets of the form $\{x \in S: x < r\}, r \in S$, and of the form $\{x \in S: x >r\}, r \in S$ which are sometimes denoted $(-\infty,r)$, or $(\leftarrow, r)$ and $(r, +\infty)$ or $(r, \rightarrow)$, respectively. So the sets denoted by Munkres are just subbasic open sets in the alternative notation. They are always well-defined in any ordered set, regardless of further structure.