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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
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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)