Suppose \(X\) is a connected space and \(f\colon X\to \mathbb{R}\) is continuous . For every two points \(p,q\in X\) every point between \(f(p)\) and \(f(q)\) is attained by \(f\).
Proof
This is a consequence of the main theorem of connectedness
and that every connected set in \(\mathbb{R}\) is an interval
.