Intermediate value theorem for connected spaces

Builds on Continuous image of a connected set is connected Connected subsets of the real line are intervals Continuous map of topological spaces

Statement

Let $X$ be a connected space, let $f \colon X \to \mathbb{R}$ be continuous, and let $a, b \in X$ . Then for each real $c$ between $f(a)$ and $f(b)$ there is $x \in X$ with $f(x) = c$ .

Remarks

Proof: $f(X)$ is connected, by continuous image of a connected set is connected. Thus it is an interval, by connected subsets of the real line are intervals, and it contains $c$ . For $X = [a, b]$ , this is the intermediate value theorem of calculus: a continuous $f \colon [a, b] \to \mathbb{R}$ takes each value between $f(a)$ and $f(b)$ . For example, each polynomial of odd degree with real coefficients has a real root.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Rational numbers object
Real numbers object
Set bounded above definition
Set definition
Ring of integers statement
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Completeness axiom statement
Cartesian product definition
Subset definition
Union of a family of sets definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Topological space definition
Complement definition
Function definition
Interval definition
Subspace topology definition
Preimage of a set definition
Metric space definition
Open ball definition
Metric topology definition
Open set definition
Closed set definition
Continuous map of topological spaces definition
Connected space definition
Connected set definition
Continuous image of a connected set is connected statement
Connected subsets of the real line are intervals statement
Intermediate value theorem for connected spaces statement

Tag	`098C0KZZAP`
URI	`m2://098C0KZZAP`
Aliases	`intermediate-value-theorem`
Kind	statement, theorem
Writing	W0 stub
MSC	54D05
Status	This copy of the site has no build data. Run `m2 status 098C0KZZAP`.
Uses	Continuous image of a connected set is connected, Connected subsets of the real line are intervals, Continuous map of topological spaces

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