State and prove the intermediate value theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $P=(a, b)$ be a point of the plane $\mathbb{R}^{2}$. Show that the set of distances from points $(x, f(x))$ on the graph of $f$ to the point $P$ is an interval $[A, \infty)$ for some value $A \geqslant 0$.