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Darboux's theorem: every derivative has the intermediate-value property
Statement
If is an interval and is differentiable, then has the intermediate value property (The intermediate value property (Darboux property) of a function on an interval: the image of every subinterval is order-convex).
Facts & Assumptions
Given: in and a real between and .
Differentiability implies continuity; the closed bounded interval is compact; and a continuous real function on a nonempty compact set attains its extrema (A function differentiable at is continuous at , Heine-Borel by bisection: every closed bounded interval is compact, Extreme value theorem: a continuous real function on a nonempty compact subset of attains a greatest and a least value).
An interior extremum of a differentiable function has derivative (Fermat's interior extremum theorem: if has a local extremum at a point interior to its domain and is differentiable at , then ).
Derivatives obey the algebra rules, and the derivative of is (Sums, scalar multiples, products and quotients: , , , and when , For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).
Proof
If or , choose that endpoint.
Suppose , and define on . Then .
If instead , apply the preceding argument to , obtaining an interior extremum of .
For sufficiently small positive , the derivative inequalities give and . Hence a minimum of on occurs at an interior point .
In either strict-order case, Fermat gives , hence . Together with the endpoint case, every intermediate value is attained.
Depends on
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- A function differentiable at $c$ is continuous at $c$
- Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact
- Extreme value theorem: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains a greatest and a least value
- Fermat's interior extremum theorem: if $f$ has a local extremum at a point $c$ interior to its domain and is differentiable at $c$, then $f'(c) = 0$
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
- The intermediate value property (Darboux property) of a function on an interval: the image of every subinterval is order-convex
Used by
- A derivative has neither a removable discontinuity nor a jump discontinuity Corollary
- An injective or monotone derivative on an interval is continuous Corollary
- The Dirichlet right-hand side gives a first-order equation with no solution Counterexample
- A differentiable function whose derivative is discontinuous Example
- False: every Henstock–Kurzweil integrable function is a derivative False statement
- Scope, endpoint, factorial, and deferred-remainder conventions Remark
Dependency tree · two levels
54 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)
- J. Lebl, Basic Analysis I, Mean value theorem (standard reference, not scraped)
- Colgate University MATH 323, Chapter 5 notes (standard reference, not scraped)