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