---- > [!theorem] Theorem. ([[intermediate value theorem]]) > Let $f:X \to Y$ be a [[continuous]] map, where $X$ is a [[connected]] [[topological space]] and $Y$ is an [[poset|ordered set]] in the [[order topology]]. If $a$ and $b$ are two points of $X$ and if $r \in Y$ lies between $f(a)$ and $f(b)$, then there exists $c \in X$ s.t. $f(c)=r$. > [!proposition] Corollary. > The intermediate value theorem of calculus is the special case wherein we take $X$ to be a closed interval in $\mathbb{R}$ and $Y$ to be $\mathbb{R}$. > [!proof]- Proof. ([[intermediate value theorem]]) > Assume the hypotheses of the theorem. The sets $A=f(X) \cap (-\infty,r) \text{ and }B=f(X)\cap(r,\infty)$ > are disjoint, and they are nonempty because one contains and $f(a)$ and the other contains $f(b)$. Each is open in $f(X)$ as the intersection of an open [[ray]] in $Y$ with $f(X)$. Since $f(X)$ is [[connected]] in $Y$ by [[continuity preserves connectedness]], $A \sqcup B=f(X) - \{ r \}$ cannot be a [[separation of a topological space|separation]] of $X$, hence $r \in f(X)$. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```