generalized intermediate value theorem

Theorem.

Let $f\mathrm{:}X\mathrm{\to }Y$ be a continuous function with $X$ a connected space and $Y$ a totally ordered set in the order topology. If $x_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{\in }X$ and $y\mathrm{\in }Y$ lies between $f\mathit{}\mathrm{(}{x}_{\mathrm{1}}\mathrm{)}$ and $f\mathit{}\mathrm{(}{x}_{\mathrm{2}}\mathrm{)}$, then there exists $x\mathrm{\in }X$ such that $f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}y$.

Proof.

The sets $U=f(X)\cap (-\mathrm{\infty },y)$ and $V=f(X)\cap (y,\mathrm{\infty })$ are disjoint open subsets of $f(X)$ in the subspace topology, and they are both non-empty, as $f(x_1)$ is contained in one and $f(x_2)$ is contained in the other. If $y\notin f(X)$, then $U\cup V$ constitutes a of the space $f(X)$, contradicting the hypothesis that $f(X)$ is the continuous image of the connected space $X$. Thus there must exist $x\in X$ such that $f(x)=y$.∎

This version of the intermediate value theorem reduces to the familiar one of http://planetmath.org/node/7599real analysis when $X$ is taken to be a closed interval in $ℝ$ and $Y$ is taken to be $ℝ$.