Intermediate value theorem for connected spaces

Statement

Let $X$ be a connected space, let $f \colon X \to \mathbb{R}$ be continuous, and let $a, b \in X$. Then for each real $c$ between $f(a)$ and $f(b)$ there is $x \in X$ with $f(x) = c$.

Remarks

Proof: $f(X)$ is connected, by continuous image of a connected set is connected. Thus it is an interval, by connected subsets of the real line are intervals, and it contains $c$. For $X = [a, b]$, this is the intermediate value theorem of calculus: a continuous $f \colon [a, b] \to \mathbb{R}$ takes each value between $f(a)$ and $f(b)$. For example, each polynomial of odd degree with real coefficients has a real root.

Explanations

L1 human

A continuous real function on a connected space cannot jump over a value. To get from $f(a)$ to $f(b)$, it passes through all values in between.