Intermediate Value Theorem

The Intermediate Value Theorem

If a continuous function moves from one value to another, it cannot skip any value in between — like a person walking from 3 ft to 6 ft who must pass through every intermediate height. The Intermediate Value Theorem (IVT) formalizes this intuition:

If $f$ is continuous on $[a,b]$, and $N$ is any number strictly between $f(a)$ and $f(b)$, then there exists at least one $c \in (a,b)$ such that $f(c) = N$.

Continuity is essential. A discontinuous function can "jump" over values instead of passing through them. For example, if $f$ jumps from $f(a)=1$ to $f(b)=5$ with a break in between, there may be no $c$ where $f(c)=3$. Continuity guarantees an unbroken graph from $(a, f(a))$ to $(b, f(b))$, so every horizontal line $y = N$ between those heights must cross the graph at least once.

Common Applications

The most frequent use of IVT is proving a root exists without finding it. Show that $f$ is continuous on $[a,b]$, then verify $f(a)$ and $f(b)$ have opposite signs. Since $0$ lies between them, the IVT guarantees some $c \in (a,b)$ with $f(c) = 0$.

Example: For $f(x) = x^3 - x - 1$ on $[1,2]$: $f(1) = -1 < 0$ and $f(2) = 5 > 0$. Since $f$ is continuous (polynomial) and changes sign, a root exists in $(1,2)$.

For target values beyond just zero, define $g(x) = f(x) - N$ and check if $g$ changes sign. If $g(c) = 0$, then $f(c) = N$.

Key Limitation

The IVT guarantees existence — at least one solution exists — but says nothing about how many solutions exist or their exact values. It's a powerful existence tool, not a solution-finding method.