Intermediate Value Theorem on Connected Sets¶

Theorem¶

Let $f\colon \mathbb{R}^n \to \mathbb{R}$ be a continuous real-valued function. Let $E$ be any connected subset of $\mathbb{R}^n\text{,}$ and let $\boldsymbol{a}$ and $\boldsymbol{b}$ be any two points in $E\text{.}$ If $y$ is a real number between $f(\boldsymbol{a})$ and $f(\boldsymbol{b})\text{,}$ then there exists a $\boldsymbol{c} \in E$ such that $f(\boldsymbol{c}) = y\text{.}$

Interpretation¶

A continuous function on a connected set cannot skip values: if it takes two different numbers, it must also take every number in between. The connected set $E$ takes over the role of the interval $[a,b]$ in the one-variable intermediate value theorem, and connectedness replaces "no gaps".

The points $\boldsymbol{a}$ and $\boldsymbol{b}$ of the connected region $E$ are sent to $f(\boldsymbol{a})$ and $f(\boldsymbol{b})$ on the real line. For the level $y$ between these values, the theorem gives a point $\boldsymbol{c}\in E$ with $f(\boldsymbol{c})=y$$y$. The curve from $\boldsymbol{a}$ to $\boldsymbol{b}$ through $\boldsymbol{c}$ shows the idea of the proof for path-connected sets in the Key Results: along a path, $f$ becomes a continuous function on $[0,1]$.

In general, the proof uses that the image $f(E)$ of a connected set is connected, and the connected subsets of $\mathbb{R}$ are the intervals. An interval that contains $f(\boldsymbol{a})$ and $f(\boldsymbol{b})$ contains every number between them.

Both hypotheses are needed. On the disconnected set $E=\{(x,y)\in\mathbb{R}^2\colon |x|\ge1\}$$1\}$, the continuous function $f(x,y)=x$ takes the values $-1$ and $1$ but never $0$. On the connected plane, the discontinuous function that equals $1$ for $x\ge0$ and $-1$ for $x<0$ also skips $0$.

A typical use is a sign change: if a continuous $f$ takes a negative and a positive value on a connected set $E$, then the level set $\{\boldsymbol{x}\in E\colon f(\boldsymbol{x})=0\}$ is nonempty. Similarly, a continuous function on a connected set that takes only integer values is constant.

Key Results¶

• Continuous image of a connected set is connected.
• If $E$ is path‑connected, the theorem can be proved by composing $f$ with a path and applying the one‑dimensional intermediate value theorem.

Examples¶

• Let $E$ be the closed unit disk in $\mathbb{R}^2$ and $f(x,y)=x^2+y^2$$y^2$​. The image is $[0,1]$ which is also a connected set.
• Let $E$ be the unit sphere in $\mathbb{R}^3$ and $f(x,y,z)=z$ (height). The image is $[-1,1]$.

Prerequisites¶

• Connected and Disconnected Sets
• Continuity between Metric Spaces

Related Concepts¶

• Intermediate Value Theorem
• Extreme Value Theorem on Compact Sets

Sources¶

• Math III: Analysis · Lecture 10

Symbols: element of, real numbers

Last updated: 20 October 2026 03:02