Theorem 2 (Bolzano)

Let $f$ be a continuous real-valued function defined on a connected space $X$. If $a$ and $b$ are any two points of $X$, and if $\alpha$ is a real number belonging to the closed interval whose end-points are $f(a)$ and $f(b)$, then there is at least one point $x \in X$ such that $f(x) = \alpha$.

For $f(X)$ is connected (Chapter I, § 11, no. 2, Proposition 4) and is therefore an interval of $\overline{\mathbf{R}}$ (§ 4, no. 2, Proposition 5); hence it contains the closed interval with end-points $f(a)$ and $f(b)$.

This theorem is often expressed in the form that a continuous real-valued function on a connected space cannot pass from one value to another without passing through all intermediate values.

This property is by no means characteristic of continuous functions; there are examples of functions defined on a connected space and discontinuous at every point which have this property (Exercise 2).