Intermediate Value Theorem¶

Theorem¶

Let $a < b$ be real numbers, and let $f\colon [a, b] \to \mathbb{R}$ be continuous. If $y$ is any number between $\min\{f(a), f(b)\}$ and $\max\{f(a), f(b)\}$, then there exists $c \in [a, b]$ such that $f(c) = y\text{.}$

Interpretation¶

The intermediate value theorem makes precise the picture that the graph of a continuous function on an interval can be drawn without lifting the pen: to get from height $f(a)$ to height $f(b)$, the graph has to pass through every height in between.

Every level $y$ between $f(a)$ and $f(b)$ is reached by the continuous $f$: here $f(c)=y$ at three points $c_1,c_2,c_3$ of $[a,b]$. The theorem promises at least one such point; it says nothing about how many there are or where they lie.

Both hypotheses are needed. Continuity rules out jumps: the signum function on $[-1,1]$ takes the values $-1$ and $1$ but never the value $\tfrac{1}{2}$. The domain must also be a whole interval without gaps: $f(x)=1/x$ is continuous on $[-1,1]$ with $0$ removed and takes the values $-1$ and $1$, but never the value $0$.

The theorem is mostly used in its sign-change form from the Examples: if $f(a)$ and $f(b)$ have opposite signs, then $f$ has a zero in $[a,b]$. It proves that an equation has a solution without solving it, and repeatedly halving the interval while keeping the sign change locates that solution numerically. Math III later extends the theorem beyond intervals.

Key Results¶

• Every value between $f(a)$ and $f(b)$ is attained on $[a,b]\text{.}$
• The theorem ensures existence but not uniqueness.
• Continuity is essential for the result.

Examples¶

• A continuous function changing sign on an interval must have a root.

Prerequisites¶

• Continuity
• Minimum of a Set
• Maximum of a Set

Related Concepts¶

• Connected Set
• Intermediate Value Theorem on Connected Sets

Sources¶

• Math I: Calculus · Lecture 5

Aliases: IVT Keywords: Bolzano's theorem Symbols: element of, real numbers

Last updated: 20 October 2026 03:02