The Intermediate Value Theorem for Continuous Functions

Theorem (Intermediate Value Theorem). If $f$ is a continuous function on a closed interval $[a,b]$, and $y_0$ is any value between $f(a)$ and $f(b)$, then $y_0=f(c)$ for some $c$ in $[a,b]$.

Geometrically (Fig. 8.10), this says that any horizontal line $y=y_0$ drawn at a height between $f(a)$ and $f(b)$ must cross the curve $y=f(x)$ at least once somewhere over the interval $[a,b]$ — the curve simply cannot skip over an in-between height without ever reaching it, precisely because it is continuous. The proof rests on the completeness property of the real number system and belongs to a more advanced course; here the theorem is used as a tool. Continuity of $f$ on the whole interval is essential — if $f$ is discontinuous at even one point of $[a,b]$, the conclusion can fail (the graph could genuinely skip a value by jumping over it). …