|
|
|
|
Darboux's theorem (analysis)
|
(Theorem)
|
|
|
Let $f:[a,b]\to\Reals$ be a real-valued continuous function on $[a,b]$ , which is differentiable on $(a,b)$ , differentiable from the right at $a$ , and differentiable from the left at $b$ . Then $f'$ satisfies the intermediate value theorem: for every $t$ between $f'_{+}(a)$ and $f'_{-}(b)$ , there is some $x\in [a,b]$
such that $f'(x)=t$ .
Note that when $f$ is continuously differentiable ($f\in C^1([a,b])$ ), this is trivially true by the intermediate value theorem. But even when $f'$ is not continuous, Darboux's theorem places a severe restriction on what it can be.
|
"Darboux's theorem (analysis)" is owned by mathwizard. [ full author list (2) | owner history (1) ]
|
|
(view preamble | get metadata)
| Other names: |
intermediate value property of the derivative |
|
|
Cross-references: restriction, places, theorem, even, continuously differentiable, intermediate value theorem, right, differentiable, continuous function
There is 1 reference to this entry.
This is version 4 of Darboux's theorem (analysis), born on 2002-06-06, modified 2004-02-25.
Object id is 3055, canonical name is DarbouxsTheorem.
Accessed 9542 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
