A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values

Let $X$ be a connected topological space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets) and let $f : X \to \mathbb{R}$ be continuous (Continuity of a map of topological spaces at a point and globally), $\mathbb{R}$ carrying its usual topology (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). Then:

1. $f[X]$ is an order-convex subset of $\mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).
2. Intermediate values are attained. If $p, q \in X$ and $c \in \mathbb{R}$ satisfies $f(p) \le c \le f(q)$, then there is $x \in X$ with $f(x) = c$.

Claim 2 is the intermediate value theorem with no hypothesis on $X$ beyond connectedness: no order, no metric, no interval. The classical statement for a continuous $f : [a,b] \to \mathbb{R}$ is the special case $X = [a,b]$, that subspace being connected by The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$".

• Why $f(p) \le c \le f(q)$ and not $f(p) < c < f(q)$. Order-convexity is stated with non-strict inequalities, so the endpoints are included and the statement covers $c = f(p)$ and $c = f(q)$ without a separate clause. No assumption $f(p) \le f(q)$ is needed either: if $f(q) \le c \le f(p)$ the same argument applies with the two points exchanged.

• What is not claimed. Nothing here says that $f[X]$ is an interval in the sense of one of the nine written forms of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length; that classification of the order-convex sets is recorded there as unproved. Nor does the corollary say anything about how many $x$ satisfy $f(x) = c$, or that such an $x$ can be found by any procedure. It is an existence statement obtained by transporting connectedness, and the witness is never exhibited.

• The hypothesis on $X$ is exactly connectedness. If $X$ is disconnected the conclusion fails at once: a separation $(U,V)$ of $X$ gives a continuous $f$ equal to $0$ on $U$ and $1$ on $V$ whose image is $\{0,1\}$, which omits every value strictly between.