Concept
Intermediate value theorem
Overview
In mathematical analysis, the intermediate value theorem states that if
f
{\displaystyle f}
is a continuous function whose domain contains the interval [a, b] and
s
{\displaystyle s}
is a number such that
f
(
a
)
<
s
<
f
(
b
)
{\displaystyle f(a)<s<f(b)}
, then there exists some
x
{\displaystyle x}
between
a
{\displaystyle a}
and
b
{\displaystyle b}
such that
f
(
x
)
=
s
{\displaystyle f(x)=s}
. That is, the image of a continuous function over an interval is itself an interval that contains
f
(
a
)
,
f
(
b
)
{\displaystyle f(a),f(b)}
.
For example, suppose that
f
∈
C
(
[
1
,
2
]
)
,
f
(
1
)
=
3
,
f
(
2
)
=
5
{\displaystyle f\in C([1,2]),f(1)=3,f(2)=5}
, then the graph of
y
=
f
(
x
)
{\displaystyle y=f(x)}
must pass through the horizontal line
y
=
4
{\displaystyle y=4}
while
x
{\displaystyle x}
moves from
1
{\displaystyle 1}
to
2
{\displaystyle 2}
. Over the interval, the set of function values has no gap, and the graph can be drawn without lifting a pencil from the paper.
The corollary Bolzano's theorem states that if a continuous function has values of opposite sign inside an interval, then it has a root in that interval. The theorem depends on, and is equivalent to, the completeness of the real numbers, although Weierstrass Nullstellensatz is a version of the intermediate value theorem for polynomials over a real closed field.
A similar result to the intermediate value theorem is the Borsuk–Ulam theorem, which underpins why rotating a wobbly table will always bring it to stability. Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property, even though they need not be continuous.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Intermediate value theorem is.
=132.= The next step, in defining a form of externality, is obtained from the idea of _dimensions_. Positions, we have seen, are defined solely by their relations to other positions. But in order that such definition may be possible, a finite number of relations must suffice, since infinite numbers are philosophically inadmissible. A position must be definable, therefore, if knowledge of our form is to be possible…
Bertrand Russell, An essay on the foundations of geometry
States what Intermediate value theorem is.
Take such a property as being named Jones. If a man is named Jones, so is his son; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his descendants in the direct male line are called Jones; this follows from the fact that the property is hereditary. Now, instead of the relation of father and son, consider the relation…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Intermediate value theorem is.
Related ideas
- If and only if
- Infinitesimal
- Function (mathematics)
- Rational number
- Squaring the circle
- Real number
- Constructivism (philosophy of mathematics)
- Set (mathematics)
- Topology
- Bernard Bolzano
- Total order
- Topological space
- Infimum and supremum
- Mathematical analysis
- Continuous function
- Keith Devlin
- Foundations of Science
- Mikhail Katz
Place in the tradition
Part of Philosophy of mathematics.
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