Intermediate value theorem

Statement

Suppose $X$ is a connected space and $f:X\to \mathbb{R}$ is a continuous map, where $\mathbb{R}$ is the real line with the usual Euclidean topology. Then, if there exist $x_1,x_2\in X$ with $f(x_1)<f(x_2)$, $f(X)$ contains the closed interval $[f(x_1),f(x_2)]$. In other words, $f$ takes all intermediate values between $f(x_1)$ and $f(x_2)$.

Related facts

• Extreme value theorem

Facts used

1. Connectedness is continuous image-closed