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An injective Darboux function on an interval is strictly monotone

Statement

An injective function $f:I\to\mathbb R$ 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 $\mathbb{R}$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)