LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-01

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An injective Darboux function on an interval is strictly monotone

Statement

An injective function $f : I \to R$ on an interval $I$ with the intermediate value property is strictly monotone.

Facts & Assumptions

Given: An injective Darboux function $f$ on an interval $I$ .

[L1]

The pointwise Darboux property in The intermediate value property (Darboux property) of a function on an interval: the image of every subinterval is order-convex attains every value between $f (a)$ and $f (b)$ on $[a, b]$ .

[L2]

Strict monotonicity has the order formulations in Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $R$ , with the dictionary to monotone sequences.

Proof

technique · contradiction

1.1

For any $a < b < c$ in $I$ , $f (b)$ must lie strictly between $f (a)$ and $f (c)$ . Indeed, if it lies above both, a value strictly between $\max {f (a), f (c)}$ and $f (b)$ is attained once in $(a, b)$ and once in $(b, c)$ , contradicting injectivity; the case below both is analogous.

assume-contraL1given

2.1

Fix $a < b$ . If $f (a) < f (b)$ , step 1.1 forces $f (x) < f (y)$ for every $x < y$ in $I$ ; inserting points between or beyond $a, b$ proves all possible placements. If $f (b) < f (a)$ , the symmetric argument gives strict decrease.

step 1.1L2cases

3.1

Injectivity excludes equality, so one of the two alternatives holds and $f$ is strictly monotone.

step 2.1discharge-contradiction∎

Depends on

Used by

An injective or monotone derivative on an interval is continuous Corollary

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
Peer-reviewed article on injective Darboux functions (DOI Serbia) (standard reference, not scraped)