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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
An injective or monotone derivative on an interval is continuous
Statement
Let be differentiable on an interval. If is injective, or if is monotone, then is continuous.
Facts & Assumptions
Given: The derivative and one of the two stated hypotheses.
Every derivative has the intermediate value property (Darboux's theorem: every derivative has the intermediate-value property).
An injective Darboux function is strictly monotone (An injective Darboux function on an interval is strictly monotone).
A nondecreasing function on an interval whose image is an interval is continuous (A function on an interval satisfying whenever , whose image is order-convex, is continuous); the increasing, decreasing, nondecreasing, and nonincreasing alternatives are those of Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of , with the dictionary to monotone sequences.
Proof
If is injective, [L1] and [L2] make it strictly monotone. If it is increasing, [L1] and [L3] make it continuous; if it is decreasing, apply [L3] to , whose interval images are the negatives of the interval images of .
If is nondecreasing, [L1] and [L3] make it continuous. If it is nonincreasing, the same argument applied to gives continuity.
These are the two stated alternatives.
Depends on
- Darboux's theorem: every derivative has the intermediate-value property
- An injective Darboux function on an interval is strictly monotone
- A function on an interval satisfying $f(x) \le f(y)$ whenever $x \le y$, whose image is order-convex, is continuous
- Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 78 results over 18 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)
- J. Lebl, Basic Analysis I, Mean value theorem (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Monotone functions (standard reference, not scraped)