In mathematics, Darboux's theorem is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conver ...
, named after
Jean Gaston Darboux
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnig ...
. It states that every function that results from the
differentiation of another function has the intermediate value property: the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of an
interval is also an interval.
When ''ƒ'' is
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in ...
(''ƒ'' in ''C''
1(
'a'',''b''), this is a consequence of the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two impo ...
. But even when ''ƒ′'' is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be.
Darboux's theorem
Let
be a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
,
be a real-valued differentiable function. Then
has the intermediate value property: If
and
are points in
with